Stochastic Differential Equations

Frederik Hendrik Kaemmerer (1839-1902) was a Dutch painter.

The topics are as follows

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Examples and Some Solution Methods.

The inclusion of random effects in differential equations leads to two distinct classes of equations, for which the solution processes have differentiable and nondifferentiable sample paths, respectively. The first
class arises when an ordinary differential equation has random coefficients, a random initial value or is forced by a fairly regular stochastic process, or when some combination of these holds. The equations are called random differential equations and are solved sample path by sample path as ordinary differential equations. The sample paths of the solution processes are then at least differentiable functions. As an example consider the linear random differential equation
\[\frac{dx}{dt}=a(\omega )x+b(t,\omega )\]
where the forcing process $b$ is continuous in $t$ for each $\omega$. For an initial value $x_{0}(\omega )$ at $t=0$, the solution is given by
\[x(t,\omega )=e^{a(\omega )t}\cdot\left (x_{0}(\omega )+\int_{0}^{t}e^{-a(\omega )s}b(s,\omega )ds\right ).\]
Its sample paths are obviously differentiable functions of $t$.

The second class occurs when the forcing is irregular stochastic process such as Gaussian white noise. The equations are then written symbolically as stochastic differentials, but are interpreted as integral equations
with Ito or Stratonovich stochastic integrals. They are called stochastic differential equations, and in general their solutions inherit the nondifferentiability of sample paths from the Brownian motions in the stochastic integral. In many applications such equations result from the incorporation of either internally or externally originating random fluctuations in the dynamical description of a system. An example of the
former is the molecular bombardment of a speck of dust on a water surface, which results in Brownian motion. The intensity of this bombardment does not depend on the state variables, for instance the position and velocity of the speck. Taking $X_{t}$ as one of the components of the velocity of the particle, Langevin wrote the equation
\begin{equation}{\label{kloeq413}}
\frac{dX_{t}}{dt}=-aX_{t}+b\xi_{t}
\end{equation}
for the acceleration of the particle. This is the sum of a retarding frictional force depending on the velocity and the molecular forces represented by a white noise process $\xi_{t}$, with intensity $b$ which is independent of the velocity. Here $a$ and $b$ are positive constants. We now interpret the Langevin equation (\ref{kloeq413}) symbolically as a stochastic differential
\begin{equation}{\label{kloeq414}}
dX_{t}=-aX_{t}dt+bdW_{t},
\end{equation}
that is as a stochastic integral equation
\begin{equation}{\label{kloeq415}}
X_{t}=X_{t_{0}}-\int_{t_{0}}^{t}aX_{s}ds+\int_{t_{0}}^{t}bdW_{s}
\end{equation}
where the second integral is an Ito stochastic integral. Similar equations arise from electrical systems where $X_{t}$ is the current and $\xi_{t}$ represents thermal noise.

With external fluctuations the intensity of the noise usually depends on the state of the system. For example, the growth coefficient in an exponential growth equation $dx/dt=\alpha x$ may fluctuate on account of
environmental effects, taking the form $\alpha =a+b\xi_{t}$, where $a$ and $b$ are positive constants and $\xi_{t}$ is a white noise process. This result in the heuristically written equation
\begin{equation}{\label{kloeq416}}
\frac{dX_{t}}{dt}=aX_{t}+bX_{t}\xi_{t}.
\end{equation}
To be precise, it is a stochastic differential
\begin{equation}{\label{kloeq417}}
dX_{t}=aX_{t}dt+bX_{t}dW_{t}
\end{equation}
or, equivalently, a stochastic integral equation
\begin{equation}{\label{kloeq418}}
X_{t}=X_{t_{0}}+\int_{t_{0}}^{t}aX_{s}ds+\int_{t_{0}}^{t}bX_{s}dW_{s}.
\end{equation}
The second integrl is again an Ito integral, but now involves the unknown solution.

In the physical sciences the random forcing in (\ref{kloeq413})-(\ref{kloeq415}) is called additive noise, whereas in (\ref{kloeq416})-(\ref{kloeq418}) is called multiplicative noise. Both cases are included in the general differential formulation
\begin{equation}{\label{kloeq419}}
dX_{t}=a(X_{t})dt+b(X_{t})dW_{t}
\end{equation}
or equivalent integral formulation
\begin{equation}{\label{kloeq4110}}
X_{t}=X_{t_{0}}+\int_{t_{0}}^{t}a(X_{s})ds+\int_{t_{0}}^{t}b(X_{s})dW_{s}
\end{equation}
for appropriate coefficients $a(x)$ and $b(x)$, which may be constants. Using Ito calculus we can verify that
\begin{equation}{\label{kloeq4111}}
X_{t}=e^{-at}\cdot X_{0}+e^{-at}\int_{0}^{t}e^{as}bdW_{s}
\end{equation}
is a solution of (\ref{kloeq413})-(\ref{kloeq415}) and that
\begin{equation}{\label{kloeq4112}}
X_{t}=X_{0}\cdot\exp\left (\left (a-\frac{b^{2}}{2}\right )\cdot t+bW_{t}\right )
\end{equation}
is a solution of (\ref{kloeq416})-(\ref{kloeq418}). We must however impose some restriction on the initial value $X_{0}$ here so the solution process $X_{t}$ is nonanticipative with respect to the Brownian motion $W_{t}$ and hence so the Ito integrals in (\ref{kloeq415}) and (\ref{kloeq418}) are meaningful. For this we need $X_{0}$ to be independent of $W_{t}$ for all $t>0$, which follows if $X_{0}$ is ${\cal F}_{0}$-measurable, where $\{{\cal F}_{t}\}_{t\geq 0}$ is the family of increasing $\sigma$-fields associated with the Brownian motion $\{W_{t}\}_{t\geq 0}$ because $X_{t}$ is then ${\cal F}_{t}$-measurable for each $t\geq 0$.

The explicit solutions (\ref{kloeq4111}) and (\ref{kloeq4112}) are the only solutions for their respective equations and given initial values in the sense that any other solution is an equivalent stochastic process, that is with the same finite dimensional probability distributions. In fact, a stronger form of uniqueness holds here: any equivalent version $\tilde{X}_{t}$ with continuous sample paths has, almost surely, the same sample paths as $X_{t}$, that is,
\[P\left\{\sup_{0\leq t\leq T}\left |\tilde{X}_{t}-X_{t}\right |>0\right\}=0\]
for any $T>0$. We then say that the solutions are pathwise unique.

In writing (\ref{kloeq4111}) and (\ref{kloeq4112}) we have assumed implicitly that we have a prescribed Brownian motion $\{W_{t}\}_{t\geq 0}$. Were we to change the Brownian motion we would again obtain a unique solution, given by the same formula with the new Brownian motion in it. We call such a solution a strong solution of the stochastic differential equation and use the term weak solution for when we are free to select a Brownian motion and then find a solution corresponding to this particular Brownian motion. Some stochastic differential equations may only have weak solutions and no strong solutions.

As with most ordinary differential equations we cannot generally find explicit formula like (\ref{kloeq4111}) and (\ref{kloeq4112}) for the solutions of stochastic differential equations and thus need to use a
numerical method to determine the solutions approximately. For this we need to know that the equation actually does have a solution, a unique solution preferably, for a given initial value. For an ordinary differential equation
\begin{equation}{\label{kloeq4113}}
\frac{dx}{dt}=a(x)
\end{equation}
this kind of information is provided by an existence and uniqueness theorem. A sufficient condition for the existence and uniqueness of a solution $x(t,x_{0})$ with initial value $x(0;x_{0})=x_{0}$ is that $a(x)$ satisfies the Lipschitz condition
\begin{equation}{\label{kloeq4114}}
|a(x)-a(y)|\leq K\cdot |x-y|
\end{equation}
for all $x,y\in \mathbb{R}$, where $K$ is a positive constant. The usual proof involves the Picard-Lindel\”{o}f method of successive approximation $\{x_{n}\}$ with
\[x_{n+1}(t)=x_{0}+\int_{0}^{t}a\left (x_{n}(s)\right )ds\]
for $n=0,1,2,\cdots$, where $x_{0}(t)=x_{0}$. The Lipschitz condition (\ref{kloeq4114}) provides a crucial inequality
\[\left |x_{n+1}(t)-x_{n}(t)\right |\leq K\cdot\int_{0}^{t}\left |x_{n}(s)-x_{n-1}(s)\right |ds.\]
This is used to establish the uniform convergence of the successive approximations to a continuous solution of the integral equation
\[x(t)=x_{0}=\int_{0}^{t}a(x(s))ds,\]
which is thus a solution of the original differential equation (\ref{kloeq4113}). The uniqueness of the solution then follows by a similar application of the Lipschitz condition. A serious deficiency is that this solution may become unbounded after a small elapse of time.

An analogous existence and uniqueness result holds for strong solutions of an stochastic differential equation provided both coefficients $a(x)$ and $b(x)$ satisfy a Lipschitz condition (\ref{kloeq4114}) and a growth bound on $a(x)$ such that
\begin{equation}{\label{kloeq4115}}
x\cdot a(x)\leq L(1+|x|^{2})\mbox{ or }|a(x)|^{2}\leq L(1+|x|^{2})
\end{equation}
for all $x\in \mathbb{R}$, where $L$ is a positive constant. Here the Brownian motion $\{W_{t}\}_{t\geq 0}$ with associated family of $\sigma$-fields $\{{\cal F}_{t}\}_{t\geq 0}$ is preassigned and the
initial value $X_{0}$ must be ${\cal F}_{0}$-measurable. The proof also uses successive approximations
\[X^{(n+1)}_{t}=X_{0}+\int_{0}^{t}a\left (X^{(n)}_{s}\right )ds+\int_{0}^{t}b\left (X^{(n)}_{s}\right )dW_{s}\]
for $n=0,1,2,\cdots$, where $X^{(0)}_{t}\equiv X_{0}$. A simple case assumes that $E[X_{0}^{2}]<\infty$ and then the growth bound is used to show that
\[\mathbb{E}\left [\sup_{0\leq t\leq T}X_{t}^{2}\right ]<\infty\]
for any fixed $0<T<\infty$. The Lipschitz condition is used in a similar way to the deterministic case to allow the mean-square convergence of the successive approximations and the mean-square uniqueness of the limiting solution. The Borel-Cantelli lemma is then applied to establish the stronger pathwise uniqueness of the soluions. Variations are possible, though technically more complicated. For example, the requirement that $\mathbb{E}[X_{0}^{2}]<\infty$ can be dropped and the Lipschitz condition (\ref{kloeq4114}) weakened to a Local Lipschitz condition
\[|a(x)-a(y)|\leq K_{N}\cdot |x-y|\]
for all $|x|,|y|\leq N$, where $K_{N}$ is a positive constant for each $N>0$. (From the mean-value theorem of calculus the latter holds for any continuously differentiable function $a(x)$). Growth bounds such as (\ref{kloeq4115}) are not required for existence or uniqueness, but their absense my result in the sample paths blowing up in a finite time, that is,
\[|X_{t}(\omega )|\rightarrow\infty\mbox{ as }t\rightarrow T(X_{0}(\omega )).\]
Here $T(X_{0}(\omega ))$ is called the {\bf explosion time}. The coefficients of the SDE
\begin{equation}{\label{kloeq4116}}
dX_{t}=-\frac{1}{2}\exp (-2X_{t})dt+\exp(-X_{t})dW_{t}
\end{equation}
do not satisfy a growth bound for $x<0$. The unique solution
\[X_{t}=\ln (W_{t}+\exp (X_{0}))\]
of (\ref{kloeq4116}) exists only for $0\leq t<T(X_{0}(\omega ))$ where
\[T(X_{0}(\omega ))=\min\{t\geq 0:W_{t}(\omega )=-\exp (X_{0}(\omega ))\}.\]

From their construction, the successive approximations $X^{(n)}_{t}$ above are obviously ${\cal F}_{t}$-measurable and have, almost surely, continuously sample paths. These properties are inherited by a limiting strong solution $X_{t}$ of the SDE (\ref{kloeq419})-(\ref{kloeq4110}). Such a solution, or more precisely the family of solutions $X_{t}^{0,x}$ with initial values $X_{0}^{0,x}=x$ a.s. for all $x\in \mathbb{R}$, is a homogeneous Markov process and is often called an Ito diffusion.

When the coefficients $a(x)$ and $b(x)$ of the SDE are sufficiently smooth the transition probabilities of this Markov process have a density $p=p(s,x;t,y)$ satisfying the Fokker-Planck equation
\[\frac{\partial p}{\partial t}+\frac{\partial}{\partial y}(ap)-\frac{1}{2}\frac{\partial^{2}}{\partial y^{2}}(\sigma p)=0\]
with $\sigma =b^{2}$.

The question of whether or not a process with a density satisfying the Fokker-Planck equation is necessarily an Ito diffusion underlies the interest in weak solutions of stochastic differential equations. While it is not true in general, an affirmative answer can be obtained when the coefficients satisfy certain basic smoothness and boundedness properties. Instead of the scalar coefficient $\sqrt{\sigma (x)}$, we could also have a vector function ${\bf b}(x)=(b_{1}(x),\cdots ,b_{k}(x))$ with $\sum_{i=1}^{k}(b_{i}(x))^{2}=\sigma (x)$ and $X_{t}$ may also be the solution of the SDE
\begin{equation}{\label{kloeq4117}}
dX_{t}=a(X_{t})dt+\sum_{i=1}^{k}b_{i}(X_{t})dW_{t}^{i}
\end{equation}
for some $k$-dimensional Brownian motion ${\bf W}_{t}=(W^{1}_{t},\cdots , W_{t}^{k})$ where $k\geq 1$. In this way the density may solve the Fokker-Planck equation of several different stochastic differential equations.

Stochastic differential equations can also be formed with more general coefficients than those in (\ref{kloeq419}). The most apparent generalization is to allow the coefficients to be nonautonomous, that is, to depend explicitly on $t$ so now we consider $a(t,x)$ and $b(t,x)$. In this case, the resulting solutions are now inhomogeneous Markov processes. The coefficients must be at least measurable in $t$, and the Lipschitz condition (\ref{kloeq4114}) and the growth condition (\ref{kloeq4115}) must hold uniformly on $0\leq t\leq T$ to ensure the existence and uniqueness of strong solutions on $0\leq t_{0}\leq t\leq T$.

Another common extension is for the coefficients to be random, that is, we consider $a(t,x,\omega )$ and $b(t,x,\omega )$. Appropriate measurability restrictions, such as ${\cal F}_{t}$-measurability, must be imposed to ensure that the integrands are nonanticipative. This situation occurs in stability analysis when we linearize about a solution $\bar{X}_{t}$ of (\ref{kloeq419}). With $Z_{t}=X_{t}-\bar{X}_{t}$ we then obtain the linear SDE
\[dZ_{t}=a'(\bar{X}_{t})\cdot Z_{t}dt+b'(\bar{X}_{t})\cdot Z_{t}dW_{t}\]
with coefficients $a(t,z,\omega )=a'(\bar{X}_{t}(\omega ))\cdot z$ and $b(t,z,\omega )=b'(\bar{X}_{t}(\omega ))\cdot z$.

We shall restrict the sttention now to stochastic differential equations
\begin{equation}{\label{kloeq4119}}
dX_{t}=a(t,X_{t})dt+b(t,X_{t})dW_{t}
\end{equation}
with nonrandom coefficients. A simple modification of the Ito formula shows that a function $Y_{t}=U(t,X_{t})$ of a strong solution of (\ref{kloeq4119}) satisfies
\[dY_{t}=\left (\frac{\partial U}{\partial t}+a\cdot\frac{\partial U}{\partial x}+\frac{1}{2}b^{2}\cdot\frac{\partial^{2}U}{\partial x^{2}}\right )dt+b\cdot\frac{\partial U}{\partial x}dW_{t},\]
for $U(t,x)$ sufficiently smooth, where the coefficients are evaluated at $(t,X_{t})$. We can use this to solve some elementary stochastic differential equations explicitly. For example, we know that
\[X_{t}=X_{0}\cdot\exp\left (W_{t}-\frac{t}{2}\right )\]
is a solution of the SDE $dX_{t}=X_{t}dW_{t}$. Then $Y_{t}=U(X_{t})=(X_{t})^{2}$ satisfies the SDE $dY_{t}=Y_{t}dt+2Y_{t}dW_{t}$, so $Y_{t}=Y_{0}\cdot\exp (2W_{t}-t)$ is a solution of this SDE.

Example. Let us consider the stochastic differential equation
\[dN_{t}=rN_{t}+\alpha N_{t}dW_{t}\mbox{ or }\frac{dN_{t}}{N_{t}}=rdt+\alpha dW_{t}.\]
Hence
\begin{equation}{\label{okseq514}}
\int_{0}^{t}\frac{dN_{s}}{N_{s}}=rt+\alpha\cdot W_{t}\mbox{ with }W_{0}=0.
\end{equation}
To evaluate the integral on the left hand side we use the Ito formula for the function $g(t,x)=\ln x$, $x>0$ and obtain
\[d(\ln N_{t})=\frac{dN_{t}}{N_{t}}+\frac{1}{2}\left (-\frac{1}{N_{t}^{2}}\right )\cdot (dN_{t})^{2}=\frac{dN_{t}}{N_{t}}-\frac{1}{2N_{t}^{2}}\cdot\alpha^{2}N_{t}^{2}dt=\frac{dN_{t}}{N_{t}}-\frac{1}{2}\alpha^{2}dt\]
with the computational rules $(dt)^{2}=0=dt\cdot dW_{t}$ and $(dW_{t})^{2}=dt$. Hence
\[\frac{dN_{t}}{N_{t}}=d(\ln N_{t})+\frac{1}{2}\alpha^{2}dt\]
or
\[\int_{0}^{t}\frac{dN_{s}}{N_{s}}=\int_{0}^{t}d(\ln N_{s})ds+\frac{1}{2}\alpha^{2}\int_{0}^{t}ds=\ln N_{t}-\ln N_{0}+\frac{1}{2}\alpha^{2}t,\]
so from (\ref{okseq514}) we conclude
\[\ln\frac{N_{t}}{N_{0}}=\left (r-\frac{1}{2}\alpha^{2}\right )t+\alpha\cdot W_{t}\]
or
\begin{equation}{\label{okseq515}}
N_{t}=N_{0}\cdot\exp\left (\left (r-\frac{1}{2}\alpha^{2}\right )t+\alpha\cdot W_{t}\right ).
\end{equation}
Next we want to show that if $W_{t}$ is independent of $N_{0}$ we should have
\[E[N_{t}]=E[N_{0}]\cdot r^{rt},\]
i.e. the same as when there is no noise. Let $Y_{t}=e^{\alpha W_{t}}$. We use the Ito formula for the function $g(t,x)=e^{\alpha x}$ and obtain
\[dY_{t}=\alpha\cdot e^{\alpha W_{t}}dW_{t}+\frac{1}{2}\alpha^{2}\cdot e^{\alpha W_{t}}dt\]
or
\[Y_{t}=Y_{0}+\alpha\int_{0}^{t}e^{\alpha W_{s}}dW_{s}+\frac{1}{2}\alpha^{2}\int_{0}^{t}e^{\alpha W_{s}}ds.\]
Since, from Proposition~\ref{okst321} (iii),
\[E\left [\int_{0}^{t}e^{\alpha W_{s}}dW_{s}\right ]=0,\]
we get
\[E[Y_{t}]=E[Y_{0}]+\frac{1}{2}\alpha^{2}\int_{0}^{t}E[Y_{s}]ds\]
or
\[\frac{d}{dt}E[Y_{t}]=\frac{1}{2}\alpha^{2}\cdot E[Y_{t}]\mbox{ with }E[Y_{0}]=1\]
or
\[E[Y_{t}]=e^{\frac{1}{2}\alpha^{2}t}.\]
From (\ref{okseq515}) and the independence of $W_{t}$ and $N_{0}$, we have
\[E[N_{t}]=e^{rt}\cdot e^{-\frac{1}{2}\alpha^{2}t}\cdot E[N_{0}]\cdot E[e^{\alpha W_{t}}]=E[N_{0}]\cdot e^{rt}.\]
Now that we have found the explicit solution $N_{t}$ in (\ref{okseq515}), we can use our knowledge about the behavior of $W_{t}$ to gain the information on these solutions. For example, we get the following:

If $r>\frac{1}{2}\alpha^{2}$ then $N_{t}\rightarrow\infty$ as $t\rightarrow\infty$ a.s.
If $r<\frac{1}{2}\alpha^{2}t$ then $N_{t}\rightarrow 0$ as $t\rightarrow\infty$ a.s.
If $r=\frac{1}{2}\alpha^{2}t$ then $N_{t}$ will fluctuate between arbitrary large and arbitrary small values as $t\rightarrow\infty$ a.s. $\sharp$

Example. Let us consider the 2-dimensional stochastic differential equation
\begin{equation}{\label{okseq519}}
d{\bf X}_{t}={\bf A}\cdot {\bf X}_{t}dt+{\bf H}_{t}dt+{\bf K}dW_{t},
\end{equation}
where $W_{t}$ is a one-domensional Brownian motion. We rewrite (\ref{okseq519}), by multiplying $\exp (-{\bf A}t)$ on both sides, as
\begin{equation}{\label{okseq5111}}
\exp (-{\bf A}t)\cdot d{\bf X}_{t}-\exp (-{\bf A}t)\cdot {\bf A}\cdot {\bf X}_{t}dt=\exp (-{\bf A}t)\cdot ({\bf H}_{t}dt+{\bf K}dW_{t}),
\end{equation}
where for a general $n\times n$ matrix ${\bf F}$, we define $\exp ({\bf F})$ to be the $n\times n$ matrix given by
\[\exp ({\bf F})=\sum_{n=0}^{\infty}\frac{1}{n!}{\bf F}^{n}.\]
Here it is tempting to relate the left hand side to $d(\exp (-{\bf A}t)\cdot {\bf X}_{t})$. To do this we use a 2-dimensional version of the Ito formula in Theorem~\ref{okst421} for the function ${\bf g}:[0,\infty )\times \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ given by
\[{\bf g}(t,x_{1},x_{2})=\exp (-{\bf A}t)\cdot\left [
\begin{array}{c}
x_{1}\\ x_{2}
\end{array}\right ],\]
we obtain that
\[d(\exp (-{\bf A}t)\cdot {\bf X}_{t})=(-{\bf A})\cdot\exp (-{\bf A}t)\cdot {\bf X}_{t}dt+\exp (-{\bf A}t)\cdot d{\bf X}_{t}.\]
Substituted in (\ref{okseq5111}) this gives
\[\exp (-{\bf A}t)\cdot {\bf X}_{t}-{\bf X}_{0}=\int_{0}^{t}\exp (-{\bf A}t)\cdot {\bf H}_{s}d+\int_{0}^{t}\exp (-{\bf A}t)\cdot {\bf K}dW_{s}\]
or
\[{\bf X}_{t}=\exp ({\bf A}t)\cdot\left ({\bf X}_{0}+\exp (-{\bf A}t)\cdot {\bf K}\cdot W_{t}+\int_{0}^{t}\exp (-{\bf A}t)\cdot ({\bf H}_{s}+{\bf A}\cdot {\bf K}\cdot W_{s})ds\right ),\]
by using integration by parts in Theorem \ref{okst415}. $\sharp$

Example. Choose $X=W$, one-dimensional Brownian motion, and
\[{\bf g}(t,x)=e^{ix}=(\cos x,\sin x)\in \mathbb{R}^{2}\mbox{ for }x\in \mathbb{R}.\]
Then
\[{\bf Y}=g(t,W)=e^{iW}=(\cos W,\sin W)\]
is by Ito’s formula again an Ito process. Its coordinates $Y_{1},Y_{2}$ satisfy
\[\left\{\begin{array}{l}
dY^{1}_{t}=-\sin W_{t}dW_{t}-\frac{1}{2}\cos W_{t}dt\\
dY^{2}_{t}=\cos W_{t}dW_{t}-\frac{1}{2}\sin W_{t}dt.
\end{array}\right .\]
Thus the process ${\bf Y}=(Y^{1},Y^{2})$, which we could call Brownian motion on the unit circle, is the solution of the stochastic differential equations
\[\left\{\begin{array}{l}
dY^{1}=-\frac{1}{2}Y^{1}dt-Y^{2}dW\\
dY^{2}=-\frac{1}{2}Y^{2}dt+Y^{1}dW.
\end{array}\right .\]
Or, in the matrix notation,
\[d{\bf Y}=-\frac{1}{2}{\bf Y}dt+{\bf K}\cdot {\bf Y}dW,\mbox{ where }{\bf K}=\left [\begin{array}{cc}
0 & -1\\ 1 & 0
\end{array}\right ]\]

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

Linear Stochastic Differential Equations.

As with linear ordinary differential equations, the general solution of a linear stochastic differential equation can be found explicitly. The method of solution also involves an integrating factor or, equivalently,
a fundamental solution of an associated homogeneous differential equation. The general form of a scalar linear stochastic differential equation is
\begin{equation}{\label{kloeq421}}
dX_{t}=(a_{1}(t)\cdot X_{t}+a_{2}(t))dt+(b_{1}(t)X_{t}+b_{2}(t))dW_{t}
\end{equation}
where the coefficients $a_{1},a_{2},b_{1},b_{2}$ are specified functions of time $t$ or constants. Provided they are Lebesgue measurable and bounded on an interval $0\leq t\leq T$, the existence and uniqueness theorem applies, ensuring the existence of a strong solution $X_{t}$ on $t_{0}\leq t\leq T$ for each $0\leq t_{0}\leq T$ and each ${\cal F}_{t_{0}}$-measurable initial value $X_{t_{0}}$ corresponding to
a given Brownian process $\{W_{t}\}_{t\geq 0}$ and associated family of $\sigma$-fields $\{{\cal F}_{t}\}_{t\geq 0}$. When the coefficients are all constants the SDE is autonomous amd its solution, which exist for all $t-t_{0}\geq 0$, are honogeneous Markov processes. In this case, it suffices to consider $t_{0}=0$. When $a_{2}(t)\equiv 0$ and $b_{2}(t)\equiv 0$ reduces to the homogeneous linear SDE
\begin{equation}{\label{kloeq422}}
dX_{t}=a_{1}(t)\cdot X_{t}dt+b_{2}\cdot X_{t}dW_{t}
\end{equation}
Obviously $X_{0}\equiv 0$ is a solution of (\ref{kloeq422}). Of far greater significance is the {\bf fundamental solution} $\Phi_{t,t_{0}}$ which satisfies the initial condition $\Phi_{t,t_{0}}=1$ since all other solutions can be expressed in terms of it. The problem is to find such a fundamental solution.

When $b_{1}(t)=0$ in (\ref{kloeq421}) the SDE has the form
\begin{equation}{\label{kloeq423}}
dX_{t}=(a_{1}(t)\cdot X_{t}+a_{2}(t))dt+b_{2}(t)dW_{t},
\end{equation}
that is the noise appears additivly. In this case we say that the SDE is linear in the narrow-sense. The homogeneous equation obtained from (\ref{kloeq423}) is then an ordinary differential equation
\[\frac{dX_{t}}{dt}=a_{1}(t)\cdot X_{t}\]
and its fundamental solution is
\[\Phi_{t,t_{0}}=\exp\left (\int_{t_{0}}^{t}a_{1}(s)ds\right ).\]
Applying the Ito formula to the transformation $U(t,x)=\Phi_{t,t_{0}}^{-1}\cdot x$ and the solution $X_{t}$ of (\ref{kloeq423}), we obtain
\begin{align*}
d\left (\Phi_{t,t_{0}}^{-1}\cdot X_{t}\right ) & =\left (\frac{d\Phi_{t,t_{0}}^{-1}}{dt}\cdot X_{t}+(a_{1}(t)\cdot X_{t}+a_{2}(t))\cdot\Phi_{t,t_{0}}^{-1}\right )dt+b_{2}(t)\cdot
\Phi^{-1}_{t,t_{0}}dW_{t}\\
& =a_{2}(t)\cdot\Phi_{t,t_{0}}^{-1}dt+b_{2}(t)\cdot\Phi_{t,t_{0}}^{-1}dW_{t}
\end{align*}
since
\[\frac{d\Phi_{t,t_{0}}^{-1}}{dt}=-\Phi_{t,t_{0}}^{-1}\cdot a_{1}(t).\]
The right hand side of
\[d\left (\Phi_{t,t_{0}}^{-1}\cdot X_{t}\right )=a_{2}(t)\cdot\Phi_{t,t_{0}}^{-1}dt+b_{2}(t)\cdot\Phi_{t,t_{0}}^{-1}dW_{t}\]
only involves known functions of $t$ and $\omega$, so can be integrated to give
\[\Phi_{t,t_{0}}^{-1}\cdot X_{t}=\Phi_{t,t_{0}}^{-1}\cdot X_{t_{0}}+\int_{t_{0}}^{t}a_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}ds+\int_{t_{0}}^{t}b_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}dW_{s}.\]
Since $\Phi_{t_{0},t_{0}}=1$ this leads to the solution
\begin{equation}{\label{kloeq425}}
X_{t}=\Phi_{t,t_{0}}\cdot\left (X_{t_{0}}+\int_{t_{0}}^{t}a_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}ds+\int_{t_{0}}^{t}b_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}dW_{s}\right )
\end{equation}
of the narrow-sense linear SDE (\ref{kloeq423}) where
\begin{equation}{\label{kloeq426}}
\Phi_{t,t_{0}}=\exp\left (\int_{t_{0}}^{t}a_{1}(s)ds\right ).
\end{equation}

The solution (\ref{kloeq425}) is a Gaussian process whenever the initial value $X_{t_{0}}$ is either a constant or a Gaussian random variable. Its mean and second order moment then both satisfy ordinary differential equations. These are stated below in (\ref{kloeq4210}) and (\ref{kloeq4211}) for the general linear SDE (\ref{kloeq421}), but the solution is now generally not Gaussian.

The general linear case is more complicated because the associated homogeneous equation (\ref{kloeq422}) is a genuine stochastic differential equation. The fundamental solution (\ref{kloeq426}) of the narrow-sense linear case satisfies the ordinary differential equation $d(\ln\Phi_{t,t_{0}})=a_{1}(t)dt$. Using this as a clue, it follows by the Ito formula that the transformed proces $\ln\Phi_{t,t_{0}}$ for the fundamental solution $\Phi_{t,t_{0}}$ of (\ref{kloeq422}) satisfies
\begin{align*}
d\left (\ln\Phi_{t,t_{0}}\right ) & =\left (a_{1}(t)\cdot\Phi_{t,t_{0}}\cdot\Phi_{t,t_{0}}^{-1}-\frac{1}{2}b_{1}^{2}(t)\cdot\Phi_{t,t_{0}}^{2}\cdot\Phi_{t,t_{0}}^{-2}\right )dt+b_{1}(t)\cdot\Phi_{t,t_{0}}\cdot\Phi_{t,t_{0}}^{-1}dW_{t}\\
& =\left (a_{1}(t)-\frac{1}{2}b_{1}^{2}(t)\right )dt+b_{1}(t)dW_{t},
\end{align*}
which consists only of known functiosn of $t$ and $\omega$. Hence
\[\ln\Phi_{t,t_{0}}=\int_{t_{0}}^{t}\left (a_{1}(s)-\frac{1}{2}b_{1}^{2}(s)\right )ds+\int_{t_{0}}^{t}b_{1}(s)dW_{s}\]
since $\Phi_{t_{0},t_{0}}=1$, or
\[\Phi_{t,t_{0}}-\exp\left (\int_{t_{0}}^{t}\left (a_{1}(s)-\frac{1}{2}b_{1}^{2}(s)\right )ds+\int_{t_{0}}^{t}b_{1}(s)dW_{s}\right ),\]
which in fact reduces to (\ref{kloeq426}) when $b_{1}(t)=0$. Similarly, applying the Ito formula to $\Phi_{t,t_{0}}^{-1}$, we obtain
\begin{equation}{\label{kloeq428}}
d\left (\Phi_{t,t_{0}}^{-1}\right )=\left (-a_{1}(t)+b_{1}^{2}(t)\right )\cdot\Phi_{t,t_{0}}^{-1}dt-b_{1}(t)\cdot\Phi_{t,t_{0}}^{-1}dW_{t}.
\end{equation}
Then, as with the narrow-sense considered above, the process $\Phi_{t,t_{0}}^{-1}\cdot X_{t}$ for a solution $X_{t}$ of the general linear equation (\ref{kloeq421}) has an explicitly integrable stochastic
differential. However, here both of the terms $\Phi_{t,t_{0}}$ and $X_{t}$ have stochastic differentials involving the same Brownian motion $W_{t}$, so the Ito formula must be used with the two component transformation $U(X_{t}^{(1)},X_{t}^{(2)})=X_{1}^{(1)}\cdot X_{t}^{(2)}$ with $X_{t}^{(1)}=\Phi_{t,t_{0}}^{-1}$ and $X_{t}^{(2)}=X_{t}$. The result is equation (\ref{kloeq3410}) with the coefficients of (\ref{kloeq421}) and (\ref{kloeq428}), that is
\begin{align*}
d\left (\Phi_{t,t_{0}}^{-1}\cdot X_{t}\right ) & =\left ((-a_{1}(t)+b_{1}^{2}(t))\cdot X_{t}+(a_{1}(t)\cdot X_{t}+a_{2}(t))\right )\Phi_{t,t_{0}}^{-1}dt\\
&-b_{1}(t)\cdot\left (b_{1}(t)\cdot X_{t}+b_{2}(t)\right )\cdot\Phi_{t,t_{0}}^{-1}dt\\
&+\left (-b_{1}(t)\cdot\Phi_{t,t_{0}}^{-1}\cdot X_{t}+(b_{1}(t)\cdot X_{t}+b_{2}(t))\cdot\Phi_{t,t_{0}}^{-1}\right )dW_{t}\\
& =\left (a_{2}(t)-b_{1}(t)\cdot b_{2}(t)\right )\cdot\Phi_{t,t_{0}}^{-1}dt+b_{2}(t)\cdot\Phi_{t,t_{0}}^{-1}dW_{t}.
\end{align*}
Integrating ad using $\Phi_{t_{0},t_{0}}=1$ we obtain
\[\Phi_{t,t_{0}}^{-1}\cdot X_{t}=X_{t_{0}}+\int_{t_{0}}^{t}\left (a_{2}(s)-b_{1}(s)\cdot b_{2}(s)\right )\cdot\Phi_{s,t_{0}}^{-1}ds+\int_{t_{0}}^{t}b_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}dW_{s}\]
and hence
\begin{equation}{\label{kloeq429}}
X_{t}=\Phi_{t,t_{0}}\cdot\left (X_{t_{0}}+\int_{t_{0}}^{t}\left (a_{2}(s)-b_{1}(s)\cdot b_{2}(s)\right )\cdot\Phi_{s,t_{0}}^{-1}ds+\int_{t_{0}}^{t}b_{2}(s)\cdot\Phi_{s,t_{0}}^{-1}dW_{s}\right )
\end{equation}
where $\Phi_{t,t_{0}}$ is given by (\ref{kloeq428}). Applying this to the linear SDE (\ref{kloeq417}), where $a_{1}(t)=a$, $b_{1}(t)=b$ and $a_{2}(t)=b_{2}(t)=0$, yields the solution (\ref{kloeq4111}). We observe that (\ref{kloeq429}) reduces to the narrow-sense solution (\ref{kloeq425}) when $b_{1}(t)=0$.

If we take the expectation of the integral form of equation (\ref{kloeq421}) and use the zero expectation property in Proposition~\ref{klol322} (ii) of an Ito integral, we obtain an ordinary differential equation for the mean $m(t)=E[X_{t}]$ of its solution, namely
\begin{equation}{\label{kloeq4210}}
\frac{dm(t)}{dt}=a_{1}(t)\cdot m(t)+a_{2}(t).
\end{equation}
We also find that the second order moment $p(t)=E[X_{t}^{2}]$ satisfies the ordinary differential equation
\begin{equation}{\label{kloeq4211}}
\frac{dp(t)}{dt}=(2a_{1}9t)+b_{1}^{2}(t))\cdot p(t)+2m(t)\cdot (a_{2}(t)+b_{1}(t)\cdot b_{2}(t))+b_{2}^{2}(t).
\end{equation}
To derive (\ref{kloeq4211}) we use the Ito formula to obtain an SDE for $X_{t}^{2}$ and then take the expectaion of the integral form of this equation. Both (\ref{kloeq4210}) and (\ref{kloeq4211}) are linear and can be solved using integrating factors. In the special case of a narrow-sense linear SDE (\ref{kloeq423}) equation (\ref{kloeq4210}) remains the same, whereas equation (\ref{kloeq4211}) simplifies to
\[\frac{dp(t)}{dt}=2a_{1}(t)\cdot p(t)+2m(t)\cdot a_{2}(t)+b_{2}^{2}(t).\]

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

Reducible Stochastic Differential Equations.

With an appropriate substitution $X_{t}=U(t,Y_{t})$ certain nonlinear stochastic differential equations
\begin{equation}{\label{kloeq431}}
dY_{t}=a(t,Y_{t})dt+b(t,Y_{t})dW_{t}
\end{equation}
can be reduced to a linear SDE in $X_{t}$
\begin{equation}{\label{kloeq432}}
dX_{t}=(a_{1}(t)\cdot X_{t}+a_{2}(t))dt+(b_{1}(t)\cdot X_{t}+b_{2}(t))dW_{t}.
\end{equation}
If $\frac{\partial U}{\partial y}(t,y)\neq 0$, the Inverse Function Theorem ensures the existence of a local inverse $y=V(t,x)$ of $x=U(t,y)$, that is with $x=U(t,V(t,x))$ and $y=V(t,U(t,y))$. A solution $Y_{t}$ of
(\ref{kloeq431}) then has the form $Y_{t}=V(t,X_{t})$ where $X_{t}$ is given by (\ref{kloeq432}) for appropriate coefficients $a_{1},a_{2},b_{1}$ and $b_{2}$. From the Ito formula
\[dU(t,Y_{t})=\left (\frac{\partial U}{\partial t}+a\cdot\frac{\partial U}{\partial y}+\frac{1}{2}b^{2}\cdot\frac{\partial^{2}U}{\partial y^{2}}\right )dt+b\cdot\frac{\partial U}{\partial y}dW_{t},\]
where the coefficients and the partial derivatives are evaluated at $(t,Y_{t})$. This coincides with a linear SDE of the form (\ref{kloeq432}) if
\begin{equation}{\label{kloeq433}}
\frac{\partial U}{\partial t}(t,y)+a(t,y)\cdot\frac{\partial U}{\partial y}(t,y)+\frac{1}{2}b^{2}(t,y)\cdot\frac{\partial^{2}U}{\partial y^{2}}(t,y)=a_{1}(t)\cdot U(t,y)+a_{2}(t)
\end{equation}
and
\begin{equation}{\label{kloeq434}}
b(t,y)\cdot\frac{\partial U}{\partial y}(t,y)=b_{1}(t)\cdot U(t,y)+b_{2}(t).
\end{equation}
Specializing to the case where $a_{1}(t)=b_{1}(t)=0$ and writing $a_{2}(t)=\alpha (t)$ and $b_{2}(t)=\beta (t)$, we obtain from (\ref{kloeq433}) the identity
\[\frac{\partial^{2}U}{\partial t\partial y}(t,y)=-\frac{\partial}{\partial y}\left (a(t,y)\cdot\frac{\partial U}{\partial y}(t,y)+\frac{1}{2}b^{2}(t,y)\cdot\frac{\partial^{2}U}{\partial y^{2}}(t,y)\right )\]
and, from (\ref{kloeq434}), the identities
\[\frac{\partial}{\partial y}\left (b(t,y)\cdot\frac{\partial U}{\partial y}(t,y)\right )=0\mbox{ and }b(t,y)\cdot\frac{\partial^{2}U}{\partial t\partial y}(t,y)+\frac{\partial b}{\partial t}(t,y)\cdot
\frac{\partial U}{\partial y}(t,y)=\beta^{\prime}(t).\]
Assume for now that $b(t,y)\neq 0$. Then, eliminating $U$ and its derivatives we obtain
\[\beta^{\prime}(t)=\beta (t)\cdot b(t,y)\cdot\left (\frac{1}{b^{2}(t,y)}\cdot\frac{\partial b}{\partial t}(t,y)-\frac{\partial}{\partial y}\left (\frac{a(t,y)}{b(t,y)}\right )+\frac{1}{2}\frac{\partial^{2}b}{\partial y^{2}}(t,y)\right ).\]
Since the left hand side is independent of $y$ this means
\[\frac{\partial\gamma}{\partial y}(t,y)=0\]
where
\[\gamma (t,y)=\frac{1}{b(t,y)}\cdot\frac{\partial b}{\partial t}(t,y)-b(t,y)\cdot\frac{\partial}{\partial y}\left (\frac{a(t,y)}{b(t,y)}-\frac{1}{2}\frac{\partial b}{\partial y}(t,y)\right ).\]
This is a sufficient condition for the reducibility of the nonlinear SDE (\ref{kloeq431}) to the explicitly integrable SDE
\begin{equation}{\label{kloeq436}}
dX_{t}=\alpha (t)dt+\beta (t)dW_{t}
\end{equation}
by means of a transformation $x=U(t,y)$. It can be determined from (\ref{kloeq433}) and (\ref{kloeq434}) which, in the special case, reduce to
\[\frac{\partial U}{\partial t}(t,y)+a(t,y)\cdot\frac{\partial U}{\partial y}(t,y)+\frac{1}{2}b^{2}(t,y)\cdot\frac{\partial^{2}U}{\partial y^{2}}(t,y)=\alpha (t)\mbox{ and }
b(t,y)\cdot\frac{\partial U}{\partial y}(t,y)=\beta (t),\]
resulting in
\[U(t,y)=C\cdot\exp\left (\int_{0}^{t}\gamma (s,y)ds\right )\cdot\int_{0}^{y}\frac{1}{b(t,z)}dz\]
where $C$ is an arbitrary constant. We remark that this method can also be used to reduce certain linear SDEs to stochastic differentials of the form (\ref{kloeq436})

A variation of the procedure is applicable to reduce a nonlinear autonomous SDE
\begin{equation}{\label{kloeq437}}
dY_{t}=a(Y_{t})dt+b(Y_{t})dW_{t}
\end{equation}
to the autonomous linear SDE
\begin{equation}{\label{kloeq438}}
dX_{t}=(a_{1}X_{t}+a_{2})dt+(b_{1}X_{t}+b_{2}0dW_{t}
\end{equation}
by means of a time-independent transformation $X_{t}=U(Y_{t})$. In this case the identities (\ref{kloeq433}) and (\ref{klowq434}) take the form
\begin{equation}{\label{kloeq439}}
a(y)\cdot\frac{dU}{dy}(y)+\frac{1}{2}b^{2}(y)\cdot\frac{d^{2}U}{dy^{2}}(y)=a_{1}\cdot U(y)+a_{2}
\end{equation}
and
\begin{equation}{\label{kloeq4310}}
b(y)\cdot\frac{dU}{dy}(y)=b_{1}\cdot U(y)+b_{2}.
\end{equation}
Assuming that $b(y)\neq 0$ and $b_{1}\neq 0$, it follows from (\ref{kloeq4310}) that
\[U(y)=C\cdot\exp (b_{1}\cdot B(y))-\frac{b_{2}}{b_{1}}\]
where
\[B(y)=\int_{y_{0}}^{y}\frac{1}{b(s)}ds\]
and $C$ is an arbitrary constant. Substituting this expression for $U(y)$ into (\ref{kloeq439}) gives
\begin{equation}{\label{kloeq4312}}
\left (b_{1}\cdot A(y)+\frac{1}{2}b_{1}^{2}-a_{1}\right )\cdot C\cdot\exp (b_{1}\cdot B(y))=a_{2}-a_{1}\cdot\frac{b_{2}}{b_{1}}
\end{equation}
where
\[A(y)=\frac{a(y)}{b(y)}-\frac{1}{2}\frac{db}{y}(y).\]
Differentiating (\ref{kloeq4312}), multiplying the result by
\[b(y)\cdot\exp (-b_{1}\cdot B(y))\cdot\frac{1}{b_{1}}\]
and then differentiating again, we obtain the relation
\begin{equation}{\label{kloeq4313}}
b_{1}\cdot\frac{dA}{dy}+\frac{d}{dy}\left (b\cdot\frac{dA}{dy}\right )=0.
\end{equation}
This is certainly satisfied if $dA/dy=0$ or if
\[\frac{d}{dy}\left (\frac{{\displaystyle \frac{d}{dy}\left (b\cdot\frac{dA}{dy}\right )}}{{\displaystyle \frac{dA}{dy}}}\right )=0\]
Provided $b_{1}$ is chosen so that
\[b_{1}=-\frac{{\displaystyle \frac{d}{dy}\left (b\cdot\frac{dA}{dy}\right )}}{{\displaystyle \frac{dA}{dy}}}.\]
Ib $b_{1}\neq 0$ its takes the form
\begin{equation}{\label{kloeq4316}}
U(y)=b_{2}\cdot B(y)+C
\end{equation}
where $b_{2}$ is chosen so that (\ref{kloeq4310}) is satisfied.

Example. For the nonlinear SDE
\[dY_{t}=-\frac{1}{2}\exp (-2Y_{t})dt+\exp (-Y_{t})dW_{t}\]
$a(y)=-\frac{1}{2}\exp (-2y)$ and $b(y)=\exp (-y)$, so $A(y)=0$. Thus (\ref{kloeq4313}) is satisfied with any $b_{1}$. For $b_{1}=0$ and $b_{2}=1$ a solution of (\ref{kloeq4310}) is $U=\exp (y)$ by (\ref{kloeq4316}). Substituting this into (\ref{kloeq439}) results in $a_{1}=a_{2}=0$. Hence $X_{t}=\exp (Y_{t})$ and (\ref{kloeq438}) reduces to the stochastic differential $dX_{t}=dW_{t}$, which has solution
\[X_{t}=W_{t}+X_{0}=W_{t}+\exp (Y_{0}).\]
The original nonlinear SDE (\ref{kloeq437}) thus has solution
\[Y_{t}=\ln (W_{t}+\exp (Y_{0})),\]
which is valid until the sample-path depedent explosion time
\[T(Y_{0}(\omega ))=\min\{t\geq 0:W_{t}(\omega )+\exp (Y_{0}(\omega ))=0\}.\]

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

Some Explicitly Solvable SDEs.

We shall list some explicitly solvable stochastic differential equations and their general solutions, which we have found in various books and papers. We classify them according to the relationship between their drift and diffusion coefficients.

Linear SDEs: Additive Noise.

Constant coefficients: homogeneous
\[dX_{t}=-\alpha\cdot X_{t}dt+\sigma dW_{t}\]
has solution
\[X_{t}=e^{-\alpha t}\left (X_{0}+\sigma\int_{0}^{t}e^{\alpha s}dW_{s}\right );\]
Constant coefficients: inhomogeneous
\[dX_{t}=(a\cdot X_{t}+b)dt+cdW_{t}\]
has solution
\[X_{t}=e^{at}\cdot\left (X_{0}+\frac{b}{a}(1-e^{-at})+c\int_{0}^{t}e^{-as}dW_{s}\right );\]
Variable coefficients:
\[dX_{t}=(a(t)\cdot X_{t}+b(t))dt+c(t)dW_{t}\]
has solution
\[X_{t}=\Phi_{t,t_{0}}\left (X_{t_{0}}+\int_{t_{0}}^{t}\Phi_{s,t_{0}}^{-1}\cdot b(s)ds+\int_{t_{0}}^{t}\Phi_{s,t_{0}}^{-1}\cdot c(s)dW_{s}\right )\]
with fundamental solution
\[\Phi_{t,t_{0}}=\exp\left (\int_{t_{0}}^{t}a(s)ds\right ).\]
For example,
\[dX_{t}=\left (\frac{2}{1+t}\cdot X_{t}+b^{2}(1+t)\right )dt+b^{2}(1+t)dW_{t}\]
has fundamental solution
\[\Phi_{t,t_{0}}=\left (\frac{1+t}{1+t_{0}}\right )^{2}\]
and general solution
\[X_{t}=\left (\frac{1+t}{1+t_{0}}\right )^{2}\cdot X_{0}+b^{2}(1+t)\cdot (W_{t}-W_{t_{0}}+t-t_{0}).\]
Usually the Brownian motion will appear in an integral, as for
\[dX_{t}=\left (\frac{b-X_{t}}{T-t}\right )dt+dW_{t}\]
which is satisfied by the process
\[X_{t}=X_{0}\left (1-\frac{t}{T}\right )+\frac{bt}{T}+(T-t)\cdot\int_{0}^{t}\frac{1}{T-s}dW_{s}\]
on the interval $0\leq t\leq T$.

Linear SDEs: Multiplicative Noise.

Constant coefficients: homogeneous
\begin{equation}{\label{kloeq446}}
dX_{t}=a\cdot X_{t}dt+b\cdot X_{t}dW_{t}
\end{equation}
has solution
\[X_{t}=X_{0}\cdot\exp\left (\left (a-\frac{b^{2}}{2}\right )\cdot t+b\cdot W_{t}\right ).\]
The two most important examples are the Ito exponential SDE
\begin{equation}{\label{kloeq447}}
dX_{t}=\frac{1}{2}X_{t}dt+X_{t}dW_{t}\mbox{ with solution }X_{t}=X_{0}\cdot\exp (W_{t})
\end{equation}
and the corresponding drift-free SDE
\[dX_{t}=X_{t}dW_{t}\mbox{ with solution }X_{t}=X_{0}\cdot\exp\left (W_{t}-\frac{t}{2}\right ).\]
Constant coefficients: inhomogeneous
\[dX_{t}=(aX_{t}+c)dt+(bX_{t}+d)dW_{t}\]
has solution
\[X_{t}=\Phi_{t}\cdot\left (X_{0}+(c-bd)\int_{0}^{t}\Phi_{s}^{-1}ds+d\int_{0}^{t}\Phi_{s}^{-1}dW_{s}\right )\]
with fundamental solution
\[\Phi_{t}=\exp\left (\left (a-\frac{b^{2}}{2}\right )\cdot t+b\cdot W_{t}\right ).\]
Variable coefficients: homogeneous
\[dX_{t}=a(t)\cdot X_{t}dt+b(t)\cdot X_{t}dW_{t}\]
has solution
\[X_{t}=X_{0}\cdot\exp\left (\int_{0}^{t}\left (a(s)-\frac{b^{2}(s)}{2}\right )ds+\int_{0}^{t}b(s)dW_{s}\right ).\]
Variable coefficients: inhomogeneous
\[dX_{t}=(a(t)\cdot X_{t}+c(t))dt+(b(t)\cdot X_{t}+d(t))dW_{t}\]
has solution
\[X_{t}=\Phi_{t,t_{0}}\left (X_{t_{0}}+\int_{t_{0}}^{t}\Phi_{s,t_{0}}^{-1}\cdot (c(s)-b(s)\cdot d(s))ds+\int_{t_{0}}^{t}\Phi_{s,t_{0}}^{-1}\cdotd(s)dW_{s}\right )\]
with fundamental solution
\[\Phi_{t,t_{0}}=\exp\left (\int_{t_{0}}^{t}\left (a(s)-\frac{b^{2}(s)}{2}\right )ds+\int_{t_{0}}^{t}b(s)dW_{s}\right ).\]

Reducible SDEs: Case I.

The Ito SDE
\begin{equation}{\label{kloeq4412}}
dX_{t}=\frac{1}{2}b(X_{t})\cdot b(X_{t})dt+b(X_{t})dW_{t}
\end{equation}
for a given differentiable function $b$ is equivalent to the Stratonovich SDE
\begin{equation}{\label{kloeq4413}}
dX_{t}=b(X_{t})\circ dW_{t}.
\end{equation}
We can either reduce the Ito SDE (\ref{kloeq4412}) to a linear SDE or integrate the Stratonovich SDE (\ref{kloeq4413}) directly to obtain the general solution
\[X_{t}=h^{-1}(W_{t}+h(X_{0}))\]
where
\begin{equation}{\label{kloeq4414}}
y=h(x)=\int_{\cdot}^{x}\frac{1}{b(s)}ds.
\end{equation}
Most standard functions of a Brownian motion satisfy SDEs of the form (\ref{kloeq4412}). Examples include the linear SDE (\ref{kloeq447}) and the follwing SDEs.
\[dX_{t}=\frac{1}{2}a(a-1)\cdot X_{t}^{1-2/a}dt+a\cdot X_{t}^{1-1/a}dW_{t}\mbox{ with solution }X_{t}=\left (W_{t}+X_{0}^{1/a}\right )^{a};\]
\[dX_{t}=\frac{1}{2}a^{2}\cdot X_{t}dt+a\cdot X_{t}dW_{t}\mbox{ with solution }X_{t}=X_{0}\cdot\exp (a\cdot W_{t});\]
\[dX_{t}=\frac{1}{2}(\ln a)^{2}\cdot X_{t}dt+(\ln a)\cdot X_{t}dW_{t}\mbox{ with solution }X_{t}=X_{0}\cdot a^{W_{t}}=X_{0}\cdot\exp\left (W_{t}\cdot\ln a\right );\]
\[dX_{t}=\frac{b^{-2X_{t}}}{2\ln b}dt+\frac{b^{-X_{t}}}{\ln b}dW_{t}\mbox{ with solution }X_{t}=\log_{b}\left (a\cdot W_{t}+b^{X_{0}}\right );\]
\[dX_{t}=-\frac{a^{2}}{2}\cdot X_{t}dt+a\cdot\sqrt{1-X_{t}^{2}}dW_{t}\mbox{ with solution }X_{t}=\sin (a\cdot W_{t}+\sin^{-1}X_{0});\]
\[dX_{t}=-\frac{a^{2}}{2}\cdot X_{t}dt-a\cdot\sqrt{1-X_{t}^{2}}dW_{t}\mbox{ with solution }X_{t}=\cos (a\cdot W_{t}+\cos^{-1}X_{0});\]
\[dX_{t}=a^{2}\cdot X_{t}\cdot (1+X_{t}^{2})dt+a\cdot (1+X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\tan (a\cdot W_{t}+\tan^{-1}X_{0});\]
\[dX_{t}=a^{2}\cdot X_{t}\cdot (1+X_{t}^{2})dt-a\cdot (1+X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\cot (a\cdot W_{t}+\cot^{-1}X_{0});\]
\[dX_{t}=\frac{1}{2}a^{2}\cdot X_{t}\cdot (2X_{t}^{2}-1)dt+a\cdot X_{t}\cdot\sqrt{X_{t}^{2}-1}dW_{t}\mbox{ with solution }X_{t}=\sec (a\cdot W_{t}+\sec^{-1}X_{0});\]
\[dX_{t}=\frac{1}{2}a^{2}\cdot X_{t}\cdot (2X_{t}^{2}-1)dt-a\cdot X_{t}\cdot\sqrt{X_{t}^{2}-1}dW_{t}\mbox{ with solution }X_{t}=\csc (a\cdot W_{t}+\csc^{-1}X_{0});\]
\[dX_{t}=\frac{a^{2}}{2}\cdot\tan X_{t}\cdot\sec^{2}X_{t}dt+a\cdot\sec X_{t}dW_{t}\mbox{ with solution }X_{t}=\sin^{-1}(a\cdot W_{t}+\sin X_{0});\]
\[dX_{t}=-\frac{a^{2}}{2}\cdot\cot X_{t}\cdot\csc^{2}X_{t}dt-a\cdot\csc X_{t}dW_{t}\mbox{ with solution }X_{t}=\cos^{-1}(a\cdot W_{t}+\cos X_{0});\]
\[dX_{t}=-a^{2}\cdot\sin X_{t}\cdot\cos^{3}X_{t}dt+a\cdot\cos^{2}X_{t}dW_{t}\mbox{ with solution }X_{t}=\tan^{-1}(a\cdot W_{t}+\tan X_{0});\]
\[dX_{t}=a^{2}\cdot\cos X_{t}\cdot\sin^{3}X_{t}dt-a\cdot\sin^{2}X_{t}dW_{t}\mbox{ with solution }X_{t}=\cot^{-1}(a\cdot W_{t}+\cot X_{0});\]
\[dX_{t}=\frac{a^{2}}{2}\cdot X_{t}dt+a\cdot\sqrt{X_{t}^{2}+1}dW_{t}\mbox{ with solution }X_{t}=\sinh (a\cdot W_{t}+\sinh^{-1}X_{0});\]
\[dX_{t}=\frac{a^{2}}{2}\cdot X_{t}dt+a\cdot\sqrt{X_{t}^{2}-1}dW_{t}\mbox{ with solution }X_{t}=\cosh (a\cdot W_{t}+\cosh^{-1}X_{0});\]
\[dX_{t}=-a^{2}\cdot X_{t}\cdot (1-X_{t}^{2})dt+a\cdot (1-X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\tanh (a\cdot W_{t}+\tanh^{-1}X_{0});\]
\[dX_{t}=a^{2}\cdot X_{t}\cdot (1-X_{t}^{2})dt-a\cdot (1-X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\coth (a\cdot W_{t}+\coth^{-1}X_{0});\]
\[dX_{t}=-\frac{a^{2}}{2}\cdot\tanh X_{t}\cdot\mbox{sech}^{2}X_{t}dt+a\cdot\mbox{sech}X_{t}dW_{t}\mbox{ with solution }X_{t}=\sinh^{-1}(a\cdot W_{t}+\sinh X_{0});\]
\[dX_{t}=-\frac{a^{2}}{2}\cdot\coth X_{t}\cdot\mbox{csch}^{2}X_{t}dt+a\cdot\mbox{csch}X_{t}dW_{t}\mbox{ with solution }X_{t}=\cosh^{-1}(a\cdot W_{t}+\cosh X_{0});\]
\[dx_{t}=a^{2}\cdot\sinh X_{t}\cdot\cosh^{3}X_{t}dt+a\cdot\cosh^{2}X_{t}dW{t}\mbox{ with solution }X_{t}=\tanh^{-1}(a\cdot W_{t}+\tanh X_{0});\]
\[dx_{t}=a^{2}\cdot\cosh X_{t}\cdot\sinh^{3}X_{t}dt-a\cdot\sinh^{2}X_{t}dW{t}\mbox{ with solution }X_{t}=\coth^{-1}(a\cdot W_{t}+\coth X_{0});\]
\[dX_{t}=dt+2\sqrt{X_{t}}dW_{t}\mbox{ with solution }X_{t}=\left (W_{t}+\sqrt{X_{0}}\right )^{2};\]
\[dX_{t}=-X_{t}\cdot (2\ln X_{t}+1)dt-2X_{t}\cdot\sqrt{-\ln X_{t}}dW_{t}\mbox{ with solution }X_{t}=\exp\left (-\left (W_{t}+\sqrt{-\ln X_{0}}\right )^{2}\right );\]
\[dX_{t}=\frac{a^{2}}{2}\cdot m\cdot X_{t}^{2m-1}dt+a\cdot X_{t}^{m}dW_{t}\mbox{ for }m\neq 1\mbox{ with solution }X_{t}=(X_{0}^{1-m}-a\cdot (m-1)\cdot W_{t})^{1/(1-m)};\]
\[dX_{t}=-\beta^{2}\cdot X_{t}\cdot (1-X_{t}^{2})dt+\beta\cdot (1-X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\frac{(1+X_{0})\cdot\exp (2\beta\cdot W_{t})+X_{0}-1}{(1+X_{0})\cdot\exp (\beta\cdot W_{t}+1-X_{0}};\]
\[dX_{t}=\frac{1}{3}X_{t}^{1/3}dt+X_{t}^{2/3}dW_{t}\mbox{ with solution }X_{t}=\left (X_{0}^{1/3}+\frac{1}{3}W_{t}\right )^{3}.\]
This last SDE has nonunique solution for $X_{0}=0$; for example,$X_{t}=0$ is also a solution.

Reducible SDEs: Case 2.

The Ito SDE
\[dX_{t}=\left (\alpha\cdot b(X_{t})+\frac{1}{2}b(X_{t})\cdot b'(X_{t})\right )dt+b(X_{t})dW_{t}\]
is equivalent to the Stratonovich SDE
\[dX_{t}=\alpha\cdot b(X_{t})dt+b(X_{t})\circ dW_{t}\]
and is reducible to the stochastic differential
\[dY_{t}=\alpha dt+dW_{t}\]
for $Y_{t}=h(X_{t})$, where $h$ is given by (\ref{kloeq4414}). Its general solution is thus
\[X_{t}=h^{-1}(\alpha t+W_{t}+h(X_{0})).\]
All of the examples in Case 1 can be modified to provide examples for this case. In particular we consider
\[dX_{t}=(1+X_{t})\cdot (1+X_{t}^{2})dt+(1+X_{t}^{2})dW_{t}\mbox{ with solution }X_{t}=\tan (t+W_{t}+\tan^{-1}X_{0});\]
\[dX_{t}=\left (\frac{1}{2}X_{t}+\sqrt{X_{t}^{2}+1}\right )dt+\sqrt{X_{t}^{2}+1}dW_{t}\mbox{ with solution }X_{t}=\sinh (t+W_{t}+\sinh^{-1}X_{0});\]
\[dX_{t}=-(\alpha +\beta^{2}\cdot X_{t})\cdot (1-X_{t}^{2})dt+\beta\cdot (1-X_{t}^{2})dW_{t}\]
with solution
\[X_{t}=\frac{(1+X_{0})\cdot\exp (-2\alpha t+2\beta\cdot W_{t})+X_{0}-1}{(1+X_{0})\cdot\exp (-2\alpha t+2\beta\cdot W_{t})+1-X_{0}}.\]

Reducible SDEs: Case 3.

The Ito SDE
\[dX_{t}=\left (\alpha\cdot b(X_{t})\cdot h(X_{t})+\frac{1}{2}b(X_{t})\cdot b'(X_{t})\right )dt+b(X_{t})dW_{t}\]
where $h$ is given by (\ref{kloeq4414}) is reducible to the Langevin SDE (\ref{kloeq414}) with $b=1$ in the variable for $Y_{t}=h(X_{t})$. Its general solution is thus
\[X_{t}=h^{-1}\left (e^{\alpha t}\cdot h(X_{0})+e^{\alpha t}\cdot\int_{0}^{t}e^{-\alpha s}dW_{s}\right ).\]
All of examples in Case 1 can be modified to provide examples for this case. In particular we consider
\[dX_{t}=-\left (\sin 2X_{t}+\frac{1}{4}\sin 4X_{t}\right )dt+\sqrt{2}\cdot\cos^{2}X_{t}dW_{t}\]
with solution
\[X_{t}=\tan^{-1}\left (e^{-t}\cdot\tan X_{0}+\sqrt{2}\cdot e^{-t}\cdot\int_{0}^{t}e^{s}dW_{s}\right );\]
\[dX_{t}=-\tanh X_{t}\cdot\left (a+\frac{b^{2}}{2}\cdot\mbox{sech}^{2}X_{t}\right )dt+b\cdot\mbox{sech}X_{t}dW_{t}\]
with solution
\[X_{t}=\sinh^{-1}\left (e^{-at}\cdot\sinh X_{0}+e^{-at}\cdot\int_{0}^{t}e^{as}dW_{s}\right ).\]

Reducible SDEs: Miscellaneous.

We shall give some examples of nonlinear reducible SDEs not included in the preceding three cases. The first is the most general form of a reducible SDE with polynomial drift of degree $n$.
\begin{equation}{\label{kloeq4450}}
dX_{t}=(a\cdot X_{t}^{n}+b\cdot X_{t})dt+c\cdot X_{t}dW_{t}
\end{equation}
with solution
\[X_{t}=\Theta_{t}\cdot\left (X_{0}^{1-n}+a\cdot (1-n)\cdot\int_{0}^{t}\Theta_{s}^{n-1}ds\right )^{1/(1-n)},\]
where
\[\Theta_{t}=\exp\left (\left (b-\frac{c^{2}}{2}\right )\cdot t+c\cdot W_{t}\right ).\]
The substitution $y=h(x)=x^{1-n}$ reduces the SDE (\ref{kloeq4450}) to a linear SDE with multiplicative noise. A special case for $n=2$ is the stochastic Verhulst equation
\[dX_{t}=(\lambda\cdot X_{t}-X_{t}^{2})dt+\sigma\cdot X_{t}dW_{t}\mbox{ with solution }X_{t}=\frac{{\displaystyle X_{0}\cdot\exp\left (\left (\lambda –
\frac{\sigma^{2}}{2}\right )\cdot t+\sigma\cdot W_{t}\right )}}{{\displaystyle 1+X_{0}\cdot\int_{0}^{t}\exp\left (\left (\lambda -\frac{\sigma^{2}}{2}\right )\cdot s+\sigma\cdot W_{s}\right )ds}};\]
For $n=3$ we have the stochastic Ginzburg-Landau equation
\[dX_{t}=\left (-X_{t}^{3}+\left (\alpha +\frac{\sigma^{2}}{2}\right )\cdot X_{t}\right )dt+\sigma\cdot X_{t}dW_{t}\mbox{ with solution }
X_{t}=\frac{X_{0}\cdot\exp (\alpha t+\sigma\cdot W_{t}}{{\displaystyle \sqrt{1+2\cdot X_{0}^{2}\cdot\int_{0}^{t}\exp(2\alpha s+2\sigma\cdot X_{t})ds}}};\]
Another example, which uses the exponential substitution $y=h(x)=\exp (-cx)$, is
\[dX_{t}=(a\cdot\exp (cX_{t})+b)dt+\sigma dW_{t}\]
with solution
\[X_{t}=X_{0}+bt+\sigma W_{t}-\frac{1}{c}\cdot\ln\left (1-ac\int_{0}^{t}\exp (cX_{0}+bcs+\sigma c\cdot W_{s})ds\right ).\]

Linear SDEs with $2$-Dimensional Noise.

As a generalization of (\ref{kloeq446}) we have
\[dX_{t}=a\cdot X_{t}dt+b_{1}\cdot X_{t}dW_{t}^{1}+b_{2}\cdot X_{t}dW_{t}^{2}\]
with solution
\[X_{t}=X_{0}\cdot\exp\left (\left (a-\frac{1}{2}\cdot\left (b_{1}^{2}+b_{2}^{2}\right )\right )\cdot t+b_{1}\cdot W_{t}^{1}+b_{2}\cdot W_{t}^{2}\right ).\]

\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}

Multidimensional Stochastic Differential Equations.

This follows from Chung and Williams \cite{chu}. We consider the stochastic differential equations of the form
\begin{equation}{\label{chueq101}}
d{\bf X}_{t}=\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}+{\bf b}({\bf X}_{t})dt,
\end{equation}
or equivalently in coordinate form
\begin{equation}{\label{chueq102}}
dX_{t}^{(i)}=\sum_{j=1}^{r}\sigma_{ij}({\bf X}_{t})dW_{t}^{(j)}+b_{i}({\bf X}_{t})dt\mbox{ for }i=1,\cdots ,d,
\end{equation}
where ${\bf W}=(W^{(1)},\cdots ,W^{(r)})$ is an $r$-dimensional Brownian motion starting from the original, and $latex \boldsymbol{\sigma}: \mathbb{R}^{d}\rightarrow \mathbb{R}^{d}\times
\mathbb{R}^{r}$ and ${\bf b}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ are Borel measurable functions. The norm of $A\in\mathbb{R}^{d}\times \mathbb{R}^{r}$ is defined by
\[\parallel A\parallel =\sqrt{\sum_{i=1}^{d}\sum_{j=1}^{r}A_{ij}^{2}}.\]
Under suitable conditions on the coefficients, solutions of (\ref{chueq101}) with initial condition running over the points in the state space generate a continuous strong Markov process, otherwise known as a
diffusion process. In this case, assuming ${\bf b}$ and $\boldsymbol{\sigma}$ are bounded, say, so that the following expectations are finite, we have
\[\mathbb{E}\left .\left [X_{t+h}^{(i)}-X_{t}^{(i)}\right |{\bf X}_{s}:0\leq s\leq t\right ]=b_{i}({\bf X}_{t})\cdot h+o(h)\]
and
\[\mathbb{E}\left .\left [(X_{t+h}^{(i)}-X_{t}^{(i)})\cdot (X_{t+h}^{(j)}-X_{t}^{(j)})\right |{\bf X}_{s}:0\leq s\leq t\right ]=a_{ij}({\bf X}_{t})\cdot h+o(h)\]
as $h\rightarrow 0$ for $i,j=1,\cdots ,d$, where ${\bf a}= \boldsymbol{\sigma}\boldsymbol{\sigma}^{T}$. The coefficient ${\bf b}$ is referred to as the {\bf drift vector}, and ${\bf a}$ is called
the diffusion matrix.

The rigorous interpretation of (\ref{chueq101}) is as a stochastic integral equation of the form
\begin{equation}{\label{chueq104}}
{\bf X}_{t}={\bf X}_{0}+\int_{0}^{t}\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}+\int_{0}^{t}{\bf b}({\bf X}_{s})ds\mbox{ for all }t\geq 0.
\end{equation}
In accordance with (\ref{chueq102}), the vector stochastic integral above is defined by
\[\left (\int_{0}^{t}\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}\right )_{i}=\sum_{j=1}^{r}\int_{0}^{t}\sigma_{ij}({\bf X}_{s})dW_{s}^{(j)}\mbox{ for }i=1,\cdots ,d.\]
Of course, the process ${\bf X}$, the coefficients $\boldsymbol{\sigma}$ and ${\bf b}$, and the initial value of ${\bf X}$, will need to satisfy certain conditions in order that the integrals in (\ref{chueq104}) are well-defined.

We consider first the case where the coefficients $\boldsymbol{\sigma}: \mathbb{R}^{d}\rightarrow \mathbb{R}^{d}\times\mathbb{R}^{r}$ and $latex {\bf b}: \mathbb{R}^{d}\rightarrow
\mathbb{R}^{d}$ are bounded and satisfy the uniform Lipschitz conditions:
\begin{equation}{\label{chueq106}}
\parallel\boldsymbol{\sigma}({\bf x})-\boldsymbol{\sigma}({\bf y})\parallel\leq K\cdot\parallel {\bf x}-{\bf y}\parallel\mbox{ and }
\parallel {\bf b}({\bf x})-{\bf b}({\bf y})\parallel\leq K\cdot\parallel {\bf x}-{\bf y}\parallel
\end{equation}
for some constant $K>0$ and all ${\bf x},{\bf y}\in \mathbb{R}^{d}$.

Theorem. (Chung and Williams \cite{chu}). There exists a continuous adapted solution ${\bf X}$ of $(\ref{chueq104})$. Moreover, its law is uniquely determined by that of ${\bf X}_{0}$ and ${\bf W}$. $\sharp$

Theorem. Suppose $\boldsymbol{\sigma}$ and ${\bf b}$ are bounded and satisfy the uniform Lipschitz conditions $(\ref{chueq106})$. Let ${\bf W}$ be a Brownian motion on a filtered probability space $(\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)$ $($a complete probability space $(\Omega ,{\cal F},P)$ together with a standard filtration $\{{\cal F}_{t}\}_{t\geq 0})$ such that ${\bf W}_{0}={\bf 0}$ $P$-a.s., and let ${\bf X}_{0}$ be an ${\cal F}_{0}$-measurable random vector. Then, there is a unique $({\cal B}\times {\cal F})$-measurable, adapted solution ${\bf X}$ of $(\ref{chueq104})$. This solution is continuous, and its law is uniquely determined by $\boldsymbol{\sigma}$, ${\bf b}$ and the laws of ${\bf X}_{0}$ and ${\bf W}$. Moreover, if ${\bf Y}$ is a $({\cal B}\times {\cal F})$-measurable adapted solution of $(\ref{chueq104})$ with the ${\cal F}_{0}$-measurable initial random vector ${\bf Y}_{0}$ in place of ${\bf X}_{0}$, then assuming ${\bf X}_{0}-{\bf Y}_{0}\in L^{2}$,
we have for all $T\geq 0$,
\[\mathbb{E}\left [\sup_{0\leq s\leq T}\parallel {\bf X}_{s}-{\bf Y}_{s}\parallel^{2}\right ]\leq 3\cdot \mathbb{E}\left [\parallel {\bf X}_{0}-{\bf Y}_{0}\parallel^{2}\right ]\cdot e^{C_{T}\cdot T}\]
for some constants $C_{T}$. $\sharp$

We now turn to the case where $\boldsymbol{\sigma}$ and ${\bf b}$ are locally Lipschitz continuous and satisfy a growth condition at infinity. Suppose $\boldsymbol{\sigma}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d} \times\mathbb{R}^{r}$ and ${\bf b}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ satisfy the conditions that for each $R>0$ there is a constant $K_{R}>0$ such that for all ${\bf x},{\bf y}\in\mathbb{R}^{d}$ satisfying $\parallel {\bf x}\parallel ,\parallel {\bf y}\parallel\leq R$,
\begin{equation}{\label{chueq1025}}
\parallel\boldsymbol{\sigma}({\bf x})-\boldsymbol{\sigma}({\bf y})\parallel\leq K_{R}\cdot\parallel {\bf x}-{\bf y}\parallel\mbox{ and }
\parallel {\bf b}({\bf x})-{\bf b}({\bf y})\parallel\leq K_{R}\cdot\parallel {\bf x}-{\bf y}\parallel,
\end{equation}
and there is $K>0$ such that for all ${\bf x}\in \mathbb{R}^{d}$
\begin{equation}{\label{chueq1027}}
\parallel\boldsymbol{\sigma}({\bf x})\parallel^{2}\leq K\cdot (1+\parallel {\bf x}\parallel^{2})
\end{equation}
and
\begin{equation}{\label{chueq1028}}
{\bf x}\cdot {\bf b}({\bf x})\leq K\cdot (1+\parallel {\bf x}\parallel^{2}).
\end{equation}
It follows from (\ref{chueq1025}) that $\boldsymbol{\sigma}$ and ${\bf b}$ are continuous. Note also that if $\parallel {\bf b}({\bf x})\parallel^{2}\leq C\cdot (1+\parallel {\bf x}\parallel^{2})$, then
\[{\bf x}\cdot {\bf b}({\bf x})\leq\frac{1}{2}\cdot\left (\parallel{\bf x}\parallel^{2}+\parallel {\bf b}({\bf x})\parallel^{2}\right )\leq\frac{1+C}{2}\cdot (1+\parallel {\bf x}\parallel^{2}).\]
Thus (\ref{chueq1028}) is weaker than the analog of (\ref{chueq1027}) for ${\bf b}$. Note also that if $\boldsymbol{\sigma}$ and ${\bf b}$ satisfy the uniform Lipschitz conditions (\ref{chueq106}), then (\ref{chueq1025}) hold.

\begin{equation}{\label{chut106}}\tag{1}\mbox{}\end{equation}

Theorem \ref{chut106}. (Chung and Williams \cite{chu}). Suppose $\boldsymbol{\sigma}$, ${\bf b}$ satisfy (\ref{chueq1025})–(\ref{chueq1028}), and ${\bf W}$ and ${\bf X}_{0}$ are as before. Then there is a unique continuous adapted solution ${\bf X}$ of $(\ref{chueq104})$. Moreover, the law of ${\bf X}$ is uniquely determined by the laws of ${\bf X}_{0}$ and ${\bf W}$. $\sharp$

Suppose $\boldsymbol{\sigma}$ and ${\bf b}$ satisfy (\ref{chueq1025}) and ${\bf W}$ is an $r$-dimensional Brownian motion starting from the origin, defined on some filtered
probability space $(\Omega ,{\cal F}, \{{\cal F}_{t}\}_{t\geq 0},P)$. For each ${\bf x}\in \mathbb{R}^{d}$, let ${\bf X}^{\bf x}$ denote the unique solution of (\ref{chueq104}) satisfying $latex {\bf
X}_{0}={\bf x}$. Note that all these solutions for varying ${\bf x}$ can be realized on a single filtered probability space on which ${\bf W}$ is a Brownian motion starting from the origin, since the deterministic initial random variable ${\bf x}$ is ${\cal F}_{0}$-measurable for all ${\bf x}\in \mathbb{R}^{d}$. By Theorem \ref{chut106}, the law of ${\bf X}^{\bf x}$ is uniquely determined by ${\bf x}$ and the law of ${\bf W}$. Thus any two realizations on different probability spaces will have the same law. A canonical realization of this law can be obtained by considering the probability measure $\mathbb{P}_{\bf x}$ induced on the space of continuous paths in $\mathbb{R}^{d}$ by ${\bf X}^{\bf x}$ and $P$. Specifically, let
\[C_{\mathbb{R}^{d}}=\{\boldsymbol{\omega}:[0,\infty )\rightarrow\mathbb{R}^{d}\mbox{ $\boldsymbol{\omega}$ is continuous}\}\]
with the natural $\sigma$-filed ${\cal M}= \sigma\{\boldsymbol{\omega}(s):0\leq s <\infty\}$ and filtration $\{{\cal M}_{t}\}_{t\geq 0}$ for ${\cal M}_{t}=\sigma\{\boldsymbol{\omega}(s):0\leq s\leq t\}$. Note that we have not completed the $\sigma$-filed nor augmented $\{{\cal M}_{t}\}_{t\geq 0}$ at this point. For each ${\bf x}\in \mathbb{R}^{d}$, let $P^{\bf x}$ denote the probability measure defined on $(C_{\mathbb{R}^{d}},{\cal M})$ by
\[P_{\bf x}(A)=P\{{\bf X}^{\bf x}(\omega )\in A\}\mbox{ for all }A\in {\cal M}.\]

The expectation with respect to $P_{\bf x}$ will be denoted by $\mathbb{E}_{\bf x}$. We will show that the canonical process $latex \boldsymbol{\omega}\mapsto \boldsymbol{\omega}(\cdot
)$, together with the probabilty measures $\{P_{\bf x}:{\bf x}\in\mathbb{R}^{d}\}$, defines a continuous strong Markov process. Let $C_{b}(\mathbb{R}^{d})$ denote the space of bounded continuous
real-valued functions on $\mathbb{R}^{d}$. For $f\in C_{b}(\mathbb{R}^{d})$, let $\parallel f\parallel =\sup_{{\bf x}\in\mathbb{R}^{d}}|f({\bf x})|$.

Theorem. Let ${\bf x}\in\mathbb{R}^{d}$, $f\in C_{b}(\mathbb{R}^{d})$, and $s,t\geq 0$. Then, we have $\mathb{P}_{\bf x}$-a.s.
\[\mathbb{E}_{\bf x}\left .\left [f(\boldsymbol{\omega}(t+s))\right |{\cal M}_{s}}\right ]=\mathbb{E}_{\boldsymbol{\omega}(s)}\left [f(\boldsymbol{\omega}(t))\right ]. \sharp\]

Theorem. For each ${\bf x}\in\mathbb{R}^{d}$, $f\in C_{b}(\mathbb{R}^{d})$, and each stopping time $\tau$ relative to $\{{\cal M}_{t}\}_{t\geq 0}$, we have for $t\geq 0$
\[\mathbb{E}_{\bf x}\left .\left [f(\boldsymbol{\omega}(\tau +t))\right |{\cal M}_{\tau +}\right ]=\mathbb{E}_{\boldsymbol{\omega}(\tau )}\left [
f(\boldsymbol{\omega}(t))\right ]\mbox{ on }\{\tau <\infty\}. \sharp\]

Now we consider the strong and weak solutions of (\ref{chueq101}).

Definition. Let $(\Omega ,{\cal F},\mathbb{P})$ be a complete probability space. Suppose that there are defined on this space a $d$-dimensional Brownian motion ${\bf W}$ that starts from the origin, and an
$\mathbb{R}^{d}$-valued random variable $\boldsymbol{\xi}$ such that $\boldsymbol{\xi}$ is independent of ${\bf W}$. Let $\{{\cal F}_{t}\}_{t\geq 0}$ be the standard filtration generated by $\boldsymbol{\xi}$ and ${\bf W}$, i.e., ${\cal F}_{t}=\sigma\{\boldsymbol{\xi},{\bf W}_{s}:0\leq s\leq t\}^{\sim}$, where $\sim$ denotes augmentation by all of the $\mathbb{P}$-null sets in ${\cal F}$. Note that ${\bf W}$ is a martingale relative to $\{{\cal F}_{t}\}_{t\geq 0}$. A strong solution of (\ref{chueq101}) with the data $(\Omega ,{\cal F},P,{\bf W},\boldsymbol{\xi})$ is a continuous $d$-dimensional process ${\bf X}$ defined on $(\Omega ,{\cal F},P)$ such that the following four conditions hold

${\bf X}$ is adapted to the filtration $\{{\cal F}_{t}\}_{t\geq 0}$
${\bf X}_{0}=\boldsymbol{\xi}$ $P$-a.s.
${\displaystyle \int_{0}^{t}\left (\parallel {\bf b}({\bf X}){s})\parallel +\parallel\boldsymbol{\sigma}({\bf X}_{s}\parallel^{2}\right )ds<\infty}$ $P$-a.s. for each $t\geq 0$.
Equation (\ref{chueq104}) holds $\mathbb{P}$-a.s. for all $t\geq 0$. $\sharp$

Since ${\bf X}$ is continuous and adapted, and $\boldsymbol{\sigma}$ and ${\bf b}$ are Borel measurable, $\boldsymbol{\sigma}({\bf X})$ and ${\bf b}({\bf X})$ are ${\cal B}\times {\cal F}$-measurable. From condition (iii) above, $\int_{0}^{t}\boldsymbol{\sigma}({\bf X}_{s}) d{\bf W}_{s}$ defines a continuous local martingale and $\int_{0}^{t} {\bf b}({\bf X}_{s})ds$ defines a continuous adapted process that is locally of bounded variation.

Definition. we say strong existence holds for the SDE $(\ref{chueq101})$ if for each quintuple $(\Omega ,{\cal F},P,{\bf W},\boldsymbol{\xi})$ satisfying the conditions of the above definition, there is a strong solution of $(\ref{chueq101})$ with this data. We say strong uniqueness holds for the SDE $(\ref{chueq101})$ if, for each possible quintuple of data, there is at most one strong solution of (\ref{chueq101}). $\sharp$

It follows from Theorem~\ref{chut106} that when $\boldsymbol{\sigma}$ and ${\bf b}$ satisfy (\ref{chueq1025}), strong existence and uniqueness hold for (\ref{chueq101}).

Example. (Ornstein-Uhlenbeck process) We consider the Ornstein-Uhlenbeck process
\begin{equation}{\label{chueq1046}}
dX_{t}=-\alpha\cdot X_{t}dt+dW_{t}.
\end{equation}
Since the coefficients $\sigma (x)=1$ and $b(x)=-\alpha x$ satisfy (\ref{chueq1025}), we have strong existence and uniqueness for this equation. Indeed, one can verify this directly as follows. Given a quintuple of data $(\Omega ,{\cal F},P,W,\xi )$, as in the above definition of a strong solution, define
\begin{equation}{\label{chueq1047}}
X_{t}=e^{-\alpha t}\cdot\xi+e^{-\alpha t}\cdot\int_{0}^{t}e^{\alpha s}dW_{s}\mbox{ for all }t\geq 0.
\end{equation}
Then $X$ is a solution of (\ref{chueq1046}), and by its definition, it is adapted to the standard filtration generated by $\xi$ and $W$. Hence $X$ is a strong solution of (\ref{chueq1046}) with the given data. Since the data was arbitrary, it follows that strong existence holds for (\ref{chueq1046}). On the other hand, for any strong solution $X$ of (\ref{chueq1046}) with associated data $(\Omega ,{\cal F},P,W,\xi )$, we have by Ito formula
\[e^{\alpha t}\cdot X_{t}-X_{0}=\int_{0}^{t}e^{\alpha s}\cdot (-\alpha\cdot X_{s})ds+\int_{0}^{t}e^{\alpha s}dW_{s}+\alpha\cdot\int_{0}^{t}e^{\alpha s}\cdot X_{s}ds=\int_{0}^{t}e^{\alpha s}dW_{s}.\]
Setting $X_{0}=\xi$ and multiplying by $e^{-\alpha t}$, we see that $X$ is of the form (\ref{chueq1047}). Thus, strong uniqueness holds. $\sharp$

Definition. Let $\mu$ be a Borel probability measure on $\mathbb{R}^{d}$. A weak solution of $(\ref{chueq101})$ with initial law $\mu$ is a sextuple $latex (\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P,{\bf
W},{\bf X})$ such that $(\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)$ is a filtered probability space, and ${\bf W}$ and ${\bf X}$ are processes defined on this space satisfying the following conditions

${\bf W}$ is a continuous $r$-dimensional Brownian motion and ${\bf X}$ is a continuous adapted $d$-dimensional process.
${\bf X}_{0}$ has distribution $\mu$.
${\displaystyle \int_{0}^{t}\left (\parallel {\bf b}({\bf X}_{s})\parallel +\parallel\boldsymbol{\sigma}({\bf X}_{s})\parallel^{2}\right )ds<\infty}$ $P$-a.s. for all $t\geq 0$.
Equation $(\ref{chueq104})$ holds $P$-a.s. for all $t\geq 0$. $\sharp$

Note that in specifying a weak solution one must give the filtered probability space and the Brownian motion, in addition to the process ${\bf X}$, to be used in (\ref{chueq104}). For brevity, we shall sometimes say ${\bf X}$ is a weak solution provided it is clear what one should use for the associated filtered probability space and Brownian motion. In (i) above, the condition that ${\bf W}$ is a martingale relative to $\{{\cal F}_{t}\}_{t\geq 0}$ can be replaced by the equivalent condition that ${\bf W}_{t+\cdot }-{\bf W}_{t}$ is independent of ${\cal F}_{t}$ for each $t\geq 0$. Clearly, any strong solution is also a weak solution, but the converse may not hold.

Definition. We say that weak existence holds for the SDE $(\ref{chueq101})$ if, for each Borel probability measure $\mu$ on $\mathbb{R}^{d}$, there is a weak solution of $(\ref{chueq101})$ with $\mu$ as its initial law. We say weak uniqueness holds for the SDE (\ref{chueq101}) if, for each Borel probability measure $\mu$ on $\mathbb{R}^{d}$, all solutions of $(\ref{chueq101})$ with $\mu$ as
initial distribution have the same law, i.e., if $(\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P,{\bf W},{\bf X})$ and $(\tilde{\Omega} ,\tilde{{\cal F}},\{\tilde{{\cal F}}_{t}\}_{t\geq 0}, \tilde{P},\tilde{{\bf W}},\tilde{{\bf X}})$ are two weak solutions of $(\ref{chueq101})$, both with initial distribution $\mu$, then $P\{{\bf X}\in A\}=\tilde{P}\{\tilde{{\bf X}}\in A\}$ for all $A\in {\cal M}$, where ${\cal M}=\sigma\{\boldsymbol{\omega}(s):0\leq s<\infty\}$ is the natural $\sigma$-field on $C_{\mathbb{R}^{d}}$. Weak uniqueness is also commonly referred to as uniqueness in law, or distributional uniqueness. $\sharp$

Example. By the last sentence of Theorem \ref{chut106}, under conditions (\ref{chueq1025})–(\ref{chueq1028}) on $\boldsymbol{\sigma}$ and ${\bf b}$, any two weak solutions of (\ref{chue104}), possibly on different filtered probability space but with the same initial distribution, are equivalent in law. Hence weak uniqueness holds for (\ref{chueq101}) under these conditions. $\sharp$

Definition. We say pathwise uniqueness holds for the SDE $(\ref{chueq101})$ if, whenever ${\bf X}$ and $\tilde{{\bf X}}$ are two weak solutions of (\ref{chueq101}) on the same filtered probability space with the same Brownian motion and the same initial random variable, we have $\mathbb{P}\{{\bf X}_{t}=\tilde{{\bf X}}_{t}\mbox{ for all }t\geq 0\}=1$, i.e., ${\bf X}$ and $\tilde{{\bf X}}$ are indistinguishable. $\sharp$

It is trivial that pathwise uniqueness implies strong uniqueness.

Proposition. Weak existence plus pathwise uniqueness implies strong existence and weak uniqueness. $\sharp$

\begin{equation}{\label{g}}\tag{G}\mbox{}\end{equation}

The Existence and Uniqueness Results.

First of all, an existence and uniqueness theorem for strong solutions of Ito stochastic differential equations will be presented. We shall do this in the slightly more general context of a non-autonomous scalar stochastic differential equation
\begin{equation}{\label{kloeq451}}
dX_{t}=a(t,X_{t})dt+b(t,X_{t})dW_{t}
\end{equation}
As mentioned earlier, (\ref{kloeq451}) is interpreted as a stochastic integral equation
\begin{equation}{\label{kloeq452}}
X_{t}=X_{t_{0}}+\int_{t_{0}}^{t}a(s,X_{s})ds+\int_{t_{0}}^{t}b(s,X_{s})dW_{s},
\end{equation}
where the first integral is a Lebesgue (or Riemann) integral for each sample path and the second integral is an Ito integral. A solution $\{X_{t}\}_{[t_{0},T]}$ of (\ref{kloeq452}) must thus have properties which
ensure that these integrals are meaningful. This holds if, for $e$ and $f$ defined by
\[e(t,\omega )=f(t,\omega )=0\mbox{ for }0\leq t\leq t_{0}\]
and
\[e(t,\omega )=a(t,X_{t}(\omega )),f(t,\omega )=b(t,X_{t}(\omega ))\mbox{ for }t_{0}\leq t\leq T,\]
the functions $\sqrt{|e|}$ and $f$ belong to the spaces ${\cal L}_{T}^{2}$ or ${\cal L}_{T}^{w}$. In turn, this follows if the coefficient functions $a$ and $b$ are sufficiently regular and if the process $X_{t}$ is regular and nonanticipative with respect to the given Brownian motions $\{W_{t}\}_{t\geq 0}$, that is ${\cal F}^{*}$-adapted where ${\cal F}^{*}=\{{\cal F}_{t}\}_{t\geq 0}$ is the family of $\sigma$-fields
associated with the Brownian motion. In addition, the integrals in (\ref{kloeq452}) should exist, at least a.s., for each $t\in [t_{0},T]$. We then call the process $\{X_{t}\}_{t_{0}\leq t\leq T}$ a solution of
(\ref{kloeq452}) on $[t_{0},T]$. For fixed coefficients $a$ and $b$, any solution $X$ will depend on the particular initial value $X_{t_{0}}$ and Brownian motion $W$ under consideration. If there is a solution for each given Brownian motion we say that the stochastic differential equation has a strong solution. Such a solution can be roughly thought of as a functional of the initial value $X_{t_{0}}$ and of the values $W_{s}$ of the Brownian mptions over the subinterval $t_{0}\leq s\leq t$. For a specified initial value $X_{t_{0}}$ the uniqueness of solutions of (\ref{kloeq451}) refers to the equivalence, $P$-almost surely, of the solution processes that satisfy the stochastic integral equation (\ref{kloeq452}). If there is a solution, then there will be a separable version which has, almost surely, continuous sample paths. We shall subsequently consider only this kind of solution. If any two solutions $X_{t}$ and $\tilde{X}_{t}$ have, almost surely, the same sample paths on $[t_{0},T]$, that is if
\[P\left\{\sup_{t_{0}\leq t\leq T}\left |X_{t}-\tilde{X}_{t}\right |>0\right\}=0,\]
we say that the solutions of (\ref{kloeq451}) are {\bf pathwise unique}.

The hypotheses of an existence and uniqueness theorem are usually sufficient, but not necessary, conditions to ensure the conclusions of the theorem. Some of those that we shall use here are quite strong, but can be weakened in several ways. In what follows the initial instant $0\leq t_{0}\leq T$ is arbitrary, but fixed, and the coefficients $a,b:[t_{0},T]\times \mathbb{R} \rightarrow \mathbb{R}$ are given. Most of the assumptions concern these coefficients.

A1 (Measurability). $a(t,x)$ and $b(t,x)$ are jointly ${\cal L}^{2}$-measurable in $(t,x)\in [t_{0},T]\times \mathbb{R}$;
A2 (Lipschitz condition). There exists a constant $K>0$ such that
\[|a(t,x)-a(t,y)|\leq K\cdot |x-y|\mbox{ and }b(t,x)-b(t,y)|\leq K\cdot |x-y|\] for all $t\in [t_{0},T]$ and $x,y\in \mathbb{R}$;
A3 (Linear growth bound). There exists a constant $K>0$ such that
\[|a(t,x)|^{2}\leq K^{2}\cdot (1+|x|^{2})\mbox{ and }|b(t,x)|^{2}\leq K^{2}\cdot (1+|x|^{2})\]
for all $t\in [t_{0},T]$ and $x,y\in \mathbb{R}$.

We shall henceforth hold fixed a Brownian motion $\{W_{t}\}_{t\geq 0}$ and an associated family of $\sigma$-fields $\{{\cal F}_{t}\}_{t\geq 0}$ for which the properties listed before are satified. The remaining assumption concerns the initial value

A4 (Initial value). $X_{t_{0}}$ is ${\cal F}_{t_{0}}$-measurable with $E[|X_{t_{0}}|^{2}]<\infty$.

The Lipschitz condition A2 provides the key estimates in both the proofs of uniqueness and of existence by the method of successive approximations. To these estimates, we shall then apply the following
useful result.

Proposition. (Kloeden and Platen \cite{klo})(The Grownwall Inequality). Let $\alpha ,\beta :[t_{0},T]\rightarrow \mathbb{R}$ be integrable with
\[0\leq\alpha (t)\leq\beta (t)+L\cdot\int_{t_{0}}^{t}\alpha (s)ds\]
for all $t\in [t_{0},T]$, where $L>0$. Then
\[\alpha (t)\leq\beta (t)+L\cdot\int_{t_{0}}^{t}e^{L\cdot (t-s)}\beta (s)ds\]
for all $t\in [t_{0},T]$. $\sharp$

Assuming that strong solutions of (\ref{kloeq451}) exist, we can show their pathwise uniqueness using just the measurability assumption A1 and the Lipschitz condition A2.

Proposition. (Kloeden and Platen \cite{klo}). If {\bf A1} and {\bf A2} hold, then the solutions of (\ref{kloeq452}) corresponding to the same initial value and the same Brownian motion are pathwise unique. $\sharp$

\begin{equation}{\label{klot453}}\tag{2}\mbox{}\end{equation}

Theorem \ref{klot453}. (Kloeden and Platen \cite{klo}). Under assumptions A1–A4 the stochastic differential equation (\ref{kloeq451}) has a pathwise unique strong solution $X_{t}$ on $[t_{0},T]$ with
\[\sup_{t_{0}\leq t\leq T}E[|X_{t}|^{2}]<\infty . \sharp\]

Variations of Theorem \ref{klot453} are possible with weakened assumptions. The most obvious is to drop the requirement that the initial value $X_{t_{0}}$ satisfies $E[|X_{t_{0}}|^{2}]<\infty$. Another obvious generalization is to replace the global Lipschitz condition {\bf A2} by a local one, that is with the condition holding with possibly different constants $K_{N}$ for $|x|,|y|\leq N$ and each $N>0$. This significantly enlarges the class of admissible coefficients since by the Mean Value Theorem every continuously differentiable function satisfies a local Lipschitz condition. In some cases it is possible to replace the Lipschitz condition on $b$ by the weaker Yamada condition; there exists an increasing function $\rho :[0,\infty )\rightarrow \mathbb{R}$ with $\rho (0)=0$ and $\int_{0+}\rho^{-2}(u)du=+\infty$ such that
\[|b(t,x)-b(t,y)|\leq\rho (|x-y|)\]
for all $x,y\in \mathbb{R}$ and $t\in [t_{0},T]$. This also implies an existence and uniqueness theorem of strong solutions.

The next result provides useful upper bounds on the higher even order moments of the solution. When the initial value is a constant it implies the existence of all such moments.

Proposition. Suppose that {\bf A1}-{\bf A4} hold and that $E[|X_{t_{0}}|^{2n}]<\infty$ for some integer $n\geq 1$. Then the solution $X_{t}$ of (\ref{kloeq451}) satisfies
\[\mathbb{E}\left [|X_{t}|^{2n}\right ]\leq\left (1+\mathbb{E}\left [|X_{t_{0}}|^{2n}\right ]\right )\cdot e^{C\cdot (t-t_{0})}\]
and
\[\mathbb{E}\left [|X_{t}-X_{t_{0}}|^{2n}\right ]\leq D\cdot\left (1+\mathbb{E}\left [|X_{t_{0}}|^{2n}\right ]\right )\cdot (t-t_{0})\cdot e^{C\cdot (t-t_{0})}\]
for all $t\in [t_{0},T]$, where $T<\infty$, $C=2n\cdot (2n+1)\cdot K^{2}$ and $D$ is a positive constant depending only on $n,K$ and $T-t_{0}$. $\sharp$

\begin{equation}{\label{okst521}}\tag{3}\mbox{}\end{equation}

Theorem \ref{okst521}. (Oksendal \cite{oks}) Let $T>0$ and ${\bf b}:[0,T]\times\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $\boldsymbol{\sigma}:[0,T]\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times m}$ be measurable functions satisfying
\begin{equation}{\label{okseq521}}
\parallel {\bf b}(t,{\bf x})\parallel +\parallel\boldsymbol{\sigma} (t,{\bf x})\parallel\leq c(1+\parallel {\bf x}\parallel );
\end{equation}
${\bf x}\in \mathbb{R}^{n}$, $t\in [0,T]$, and for some constant $c$, where $\parallel\boldsymbol{\sigma}\parallel =\sum|\sigma_{ij}|^{2}$ and such that
\begin{equation}{\label{okseq522}}
\parallel {\bf b}(t,{\bf x})-{\bf b}(t,{\bf y})\parallel +\parallel\boldsymbol{\sigma}(t,{\bf x})-\boldsymbol{\sigma}(t,{\bf y})\parallel\leq d\cdot\parallel {\bf x}-{\bf y}\parallel
\end{equation}
${\bf x},{\bf y}\in \mathbb{R}^{n}$, $t\in [0,T]$, and for some constant $d$. Let $Z$ be a random variable which is independent of the $\sigma$-algebra ${\cal F}_{\infty}^{(m)}$ generated by $latex {\bf
W}_{s}$, $s\geq 0$ and such that $E[|Z|^{2}]<\infty$. Then the stochastic differential equation
\begin{equation}{\label{okseq523}}
d{\bf X}_{t}={\bf b}(t,{\bf X}_{t})dt+\boldsymbol{\sigma}(t,{\bf X}_{t})d{\bf W}_{t}\mbox{ for }0\leq t\leq T, {\bf X}_{0}=Z
\end{equation}
has a unique $t$-continuous solution ${\bf X}_{t}$ with the property that
\begin{equation}{\label{okseq524}}
\mbox{${\bf X}_{t}$ is adapted to the filtration ${\cal F}_{t}^{Z}$ generated by $Z$ and ${\bf W}_{s}$ for $s\leq t$}
\end{equation}
and
\begin{equation}{\label{okseq525}}
\mathbb{E}\left [\int_{0}^{T}\parallel {\bf X}_{t}\parallel^{2}dt\right ]<\infty . \sharp
\end{equation}

Conditions (\ref{okseq521}) and (\ref{okseq522}) are natural in view of the following simple examples from deterministic differential equations (i.e. $\boldsymbol{\sigma}=0$). The equation
\[\frac{dX_{t}}{dt}=(X_{t})^{2}\mbox{ with }X_{0}=1\]
corresponding to $b(x)=x^{2}$, which does not satisft (\ref{okseq521}), has the (unique) solution
\[X_{t}=\frac{1}{1-t}, 0\leq t<1.\]
Thus it is impossible to find a global solution defined for all $t$ in this case. More generally, condition (\ref{okseq521}) ensures that the solution ${\bf X}_{t}$ of (\ref{okseq523}) does not explode, i.e., that $|{\bf X}_{t}|$ does not tend to $\infty$ in a finite time. The equation
\begin{equation}{\label{okseq527}}
\frac{dX_{t}}{dt}=3X_{t}^{2/3}\mbox{ with }X_{0}=0
\end{equation}
has more than one solution. In fact, for any $a>0$, the function
\[X_{t}=\left\{\begin{array}{ll}
0 & \mbox{for $t\leq a$}\\
(t-a)^{3} & \mbox{$t>a$}
\end{array}\right .\]
solves (\ref{okseq527}). In this case $b(x)=3x^{2/3}$ does not satisfy the Lipschitz condition (\ref{okseq522}) at $x=0$. Thus condition (\ref{okseq522}) guarantees that equation (\ref{okseq523}) has a unique
solution. Here uniqueness means that if $X^{1}_{t}(\omega )$ and $X^{2}_{t}(\omega )$ are two $t$-continuous processes satisfying (\ref{okseq523}), (\ref{okseq524}) and (\ref{okseq525}) then
\begin{equation}{\label{okseq528}}
\mbox{$X^{1}_{t}(\omega )=X^{2}_{t}(\omega )$ for all $t\leq T$ a.s.}
\end{equation}

The solution ${\bf X}_{t}$ found above is called a strong solution, because the version ${\bf W}_{t}$ of Borwinan motion is given in advance and the solution ${\bf X}_{t}$ constructed from it is ${\cal F}_{t}^{Z}$-adapted. If we are only given the functions ${\bf b}(t,{\bf x})$ and $\boldsymbol{\sigma}(t,{\bf x})$ and ask for a pair of processes $latex ((\tilde{{\bf X}}_{t},\tilde{{\bf W}}_{t}),
{\cal H}_{t})$ on a probability space $(\Omega ,{\cal H},P)$ such that (\ref{okseq523}) holds, then the solution $\tilde{{\bf X}}_{t}$, or more precisely $(\tilde{{\bf X}}_{t},\tilde{{\bf W}}_{t})$, is called a
weak solution. Here ${\cal H}_{t}$ is an increasing family of $\sigma$-fileds such that $\tilde{{\bf X}}_{t}$ is ${\cal H}_{t}$-adapted and $\tilde{{\bf W}}_{t}$ is an ${\cal H}_{t}$-Brownain motion, i.e.,
$\tilde{{\bf W}}_{t}$ is a Brownian motion, and $\tilde{{\bf W}}_{t}$ is a martingale with respect to ${\cal H}_{t}$. (and so $E[\tilde{{\bf W}}_{t}-\tilde{{\bf W}}_{t}|{\cal H}_{t}]=0$ for all $t,h\geq 0$). This allows us to define the Ito integral on the right hand side of (\ref{okseq523}) exactly as before, even though $\tilde{{\bf X}}_{t}$ need not be ${\cal F}_{t}^{Z}$-adapted. A strong solution is of course also a weak solution, but the converse is not true in general.

The uniqueness (\ref{okseq528}) that we obtain above is called strong uniqueness, while weak uniqueness simply means that any two solution (weak or strong) are identical in law, i.e., have the same
finite-dimensional distributions.

\begin{equation}{\label{oksl531}}\tag{4}\mbox{}\end{equation}

Proposition \ref{oksl531}. (Oksendal \cite{oks}). If ${\bf b}$ and $\boldsymbol{\sigma}$ satisfy the conditions of Theorem \ref{okst521}, then we have: A solution (weak or strong) of
(\ref{okseq523}) is weakly unique. $\sharp$

The concept is convenient for mathematical reasons, because there are stochastic differential equations which have no strong solutions but still a (weakly) unique weak solution. Here is a simple example.

Example. (The Tanaka Equation). Consider the one-dimensional stochastic differential equation
\begin{equation}{\label{okseq531}}
dX_{t}=sign(X_{t})dW_{t}\mbox{ with }X_{0}=0,
\end{equation}
where
\[sign(x)=\left\{\begin{array}{ll}
+1 & \mbox{if $x\geq 0$}\\ -1 & \mbox{if $x<0$}.
\end{array}\right .\]
Note that $\sigma (t,x)=\sigma (x)=sign (x)$ does not satisfy the Lipschitz condition (\ref{okseq522}), so Theorem~\ref{okst521} does not apply. Indeed, the equation (\ref{okseq531}) has no strong solution. To see this, let $\widehat{W}_{t}$ be a Brownian motion generating the filtration
$\widehat{{\cal F}}_{t}$ and define
\[Y_{t}=\int_{0}^{t}sign(\widehat{W}_{s})d\widehat{W}_{s}.\]
By the Tanaka formula \cite{okseq4312} we have
\[Y_{t}=|\widehat{W}_{t}|-|\widehat{W}_{0}|-\widehat{L}_{t}(\omega ),\]
where $\widehat{L}_{t}(\omega )$ is the local time for $\widehat{W}_{t} (\omega $latex at $0$. It follows that $Y_{t}$ is measurable with respect to the $\sigma$-field ${\cal G}_{t}$ generated by $|\widehat{W}_{s}|$ for $s\leq t$, which is clearly strictly contained in $\widehat{{\cal F}}_{t}$. Hence the $\sigma$-field ${\cal N}_{t}$ generated by $Y_{s}$ for $s\leq t$ is also strictly contained in $\widehat{{\cal F}}_{t}$. Now suppose $X_{t}$ is a strong solution of (\ref{okseq531}). Then by Theorem~\ref{okst842} it follows that $X_{t}$ is a Brownian motion with respect to the measure $P$. Let ${\cal M}_{t}$ be the $\sigma$-field generated by $X_{s}$ for $s\leq t$. Since $(sign(x))^{2}=1$ we can rewrite (\ref{okseq531}) as
\[dW_{t}=sign(X_{t})dX_{t}.\]
By the above argument applied to $\widehat{W}_{t}=X_{t}$, $Y_{t}=W_{t}$ we conclude that ${\cal F}_{t}$ is strictly contained in ${\cal M}_{t}$. But this contradicts that $X_{t}$ is a strong solution. Hence strong solutions of (\ref{okseq531}) do not exist. To find the weak solution of (\ref{okseq531}) we simply choose $X_{t}$ to be any Brownian motion $\widehat{W}_{t}$. Then we define $\widetilde{W}_{t}$ by
\[\widetilde{W}_{t}=\int_{0}^{t}sign(\widehat{W}_{s})d\widehat{W}_{s}=\int_{0}^{t}sign(X_{s})dX_{s}\]
that is,
\[d\widetilde{W}_{t}=sign(X_{t})dX_{t},\]
Then
\[dX_{t}=sign(X_{t})d\widetilde{W}_{t},\]
so $X_{t}$ is a weak solution. Finally, weak uniqueness follows from Theorem \ref{okst842}, which implies that any weak solution $X_{t}$ must be a Brownian motion with respect to $\mathbb{P}$. $\sharp$

\begin{equation}{\label{g}}\tag{G}\mbox{}\end{equation}

Diffusion Theory.

Diffusion Processes.

Markov processes taking continuous values in $\mathbb{R}$ require a somewhat more complicated mathematical framework than their discrete state counterparts, especially when they also involve continuous
time values. A rich and useful class of such Markov processes are the diffusion processes. In what follows we shall always suppose that for $k=1,2,\cdots$ every joint distribution $F_{t_{1},t_{2},\cdot ,t_{k}}(x_{1},x_{2}, \cdots ,x_{k})$ of the process $\{X_{t}\}_{t\geq 0}$ under consideration has a density $f(t_{1},x_{1};t_{2},x_{2};\cdots ;t_{k},x_{k})$. Then we define the conditional probabilities
\begin{equation}{\label{kloeq171}}
\mathbb{P}\{X_{t_{n+1}}\in B|X_{t_{1}}=x_{1},\cdots ,X_{t_{n}}=x_{n}\}=\frac{{\displaystyle \int_{B}f(t_{1},x_{1};\cdots ;t_{n},x_{n};t_{n+1},y)dy}}{{\displaystyle \int_{-\infty}^{\infty}f(t_{1},x_{1};\cdots ;t_{n},x_{n};t_{n+1},y)dy}}
\end{equation}
for all Borel subsets $B$ of $\mathbb{R}$, time instants $0<t_{1}<t_{2}< \cdots <t_{n}<t_{n+1}$ and all states $x_{1},x_{2},\cdots ,x_{n}\in \mathbb{R}$, provided the denominators are nonzero (When the denominator in (\ref{kloeq171}) is zero we either define the conditional probability to be zero or we leave it undefined). In this context the Markov property takes the form
\begin{equation}{\label{kloeq172}}
\mathbb{P}\{X_{t_{n+1}}\in B|X_{t_{1}}=x_{1},\cdots ,X_{t_{n}}=x_{n}\}=\mathbb{P}\{X_{t_{n+1}}\in B|X_{t_{n}}=x_{n}\}
\end{equation}
for all Borel subsets $B$ of $\mathbb{R}$, time instants $0<t_{1}<t_{2}< \cdots <t_{n}<t_{n+1}$ and all states $x_{1},x_{2},\cdots ,x_{n}\in \mathbb{R}$ for which the conditional probabilities are defined. If (\ref{kloeq172}) is satisfied we call the stochastic process $X_{t}$ a Markov process and write its transition probabilities as
\[P(s,x;t,B)=P\{X_{t}\in B|X_{s}=x\},\]
where $s<t$. For fixed $s,x$ and $t$, $P(s,x;t,B)$ is a probability function (measure) on the $\sigma$-field ${\cal B}$ of Borel subsets of $\mathbb{R}$. Under our assumption above it has a density $p(s,x;t,\cdot )$, called a transition density, so
\[P(s,x,;t,B)=\int_{B}p(s,x,;t,y)dy\]
for all $B\in {\cal B}$. For techincal convenience we usually also define $P(s,x;s,B)=I_{B}(x)$ for $t=s$, where $I_{B}$ is the indicator function of $B$. We say that a Markov process is homogeneous if all of its transition densities $p(s,x;t,y)$ depend only on the time difference $t-s$ rather than on the specific values of $s$ and $t$. Example of homogeneous Markov processes are standard Brownian motion with transition density
\[p(s,x;t,y)=\frac{1}{\sqrt{2\pi\cdot (t-s)}}\cdot\exp\left (-\frac{(y-x)^{2}}{2(t-s)}\right ).\]
Since $p(s,x;t,y)=p(0,x;t-s,y)$ we usually omit the superfluous first variable and simply write $p(x;t-s,y)$ with $P(x;t-s,B)=P(0,x;t-s,B)$ for the transition probabilities. The Chapman-Kolmogorov equation is
\begin{equation}{\label{kloeq175}}
p(s,x;t,y)=\int_{-\infty}^{\infty}p(s,x;\tau ,z)\cdot p(\tau ,z;t,y)dz
\end{equation}
for all $s\leq\tau\leq t$ and $x,y\in \mathbb{R}$. For transition probabilities it takes the form
\[P(s,x;t,B)=\int_{-\infty}^{\infty}P(\tau ,z;t,B)P(s,x;\tau ,dz),\]
where $B\in {\cal B}$ and the integral is an improper Riemann-Stieltjes integral.

A Markov process with transition densities $p(s,x;t,y)$ is called a diffusion process if the following three limits exist for all $\epsilon >0$, $s\geq 0$ and $x\in \mathbb{R}$

\begin{equation}{\label{kloeq179}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{|y-x|>\epsilon}p(s,x;t,y)dy=0
\end{equation}
\begin{equation}{\label{kloeq1710}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{|y-x|<\epsilon}(y-x)\cdot p(s,x;t,y)dy=a(s,x)
\end{equation}
and
\begin{equation}{\label{kloeq1711}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{|y-x|<\epsilon}(y-x)^{2}\cdot p(s,x;t,y)dy=b^{2}(s,x),
\end{equation}
where $a$ and $b$ are well-defined functions. Condition (\ref{kloeq179}) prevents a diffusion process from having instantaneous jumps. The quantity $a(s,x)$ is called the drift of the diffusion process and $b(s,x)$ its {\bf diffusion coefficient} at time $s$ and position $x$. (\ref{kloeq1710}) implies that
\[a(s,x)=\lim_{t\downarrow s}\frac{1}{t-s}\cdot E[X_{t}-X_{s}|X_{s}=x],\]
so the drift $a(s,x)$ is the instantaneous rate of change in the mean of the process given that $X_{s}=x$. Similarly, it follows from (\ref{kloeq1711}) that the squared diffusion coefficients
\[b^{2}(s,x)=\lim_{t\downarrow s}\frac{1}{t-s}\cdot \mathbb{E}[(X_{t}-X_{s})^{2}|X_{s}=x]\]
denotes the instataneous rate of change of the squared fluctuation of the process given that $X_{s}=x$.

When the drift $a$ and diffusion coefficient $b$ of a diffusion process are moderately smooth functions, then its transition density $p(s,x;t,y)$ also satisfies partial differential equations. These are the
Kolmogorov forward equation
\begin{equation}{\label{kloeq1714}}
\frac{\partial p}{\partial t}+\frac{\partial}{\partial y}(p\cdot a(t,y))-\frac{1}{2}\cdot\frac{\partial^{2}}{\partial y^{2}}(p\cdot b^{2}(t,y))=0\mbox{ for }(s,x)\mbox{ fixed},
\end{equation}
and the Kolmogorov backward equation
\begin{equation}{\label{kloeq1715}}
\frac{\partial p}{\partial s}+a(s,x)\cdot\frac{\partial p}{\partial x}+\frac{1}{2}\cdot b^{2}(s,x)\cdot\frac{\partial^{2}p}{\partial x^{2}}=0\mbox{ for $(t,y)$ fixed},
\end{equation}
with the former giving the forward evolution with respect to the final state $(t,y)$ and the latter giving the backward evolution with respect to the initial state $(s,x)$. The forward equation (\ref{kloeq1714}) is
the formal adjoint of the backward equation (\ref{kloeq1715}) and is commonly called the {\bf Fokker-Planck equation}. Both follow from the Chapman-Kolmogorov equation (\ref{kloeq175}) and (\ref{kloeq179})–(\ref{kloeq1711}).

For a stationary diffusion process there usually exists a stationary probability density $\bar{p}(y)$ such that
\[\bar{p}(y)=\int_{-\infty}^{\infty}p(s,x;t,y)\cdot\bar{p}(x)dx\]
for all $0\leq s\leq t$ and $y\in \mathbb{R}$. This density $\bar{p}$ then satisfies the corresponding stationary or time-independent Fokker-Planck equation, which, in this one-dimensional case, is the ordinary differential equation
\[\frac{d}{dy}(a(y)\cdot\bar{p}(y))-\frac{1}{2}\cdot\frac{d^{2}}{dy^{2}}(b^{2}(y)\cdot\bar{p}(y))=0\]

with drift $a$ and diffusion coefficient $b$ independent of $t$. Naturally $\bar{p}(y)\geq 0$ for all $y\in \mathbb{R}$ and $\int_{-\infty}^{\infty} \bar{p}(y)=1$. Such a diffusion process $\{X_{t}\}_{t\geq 0}$ is said to be ergodic if the following time average limit exists and equals the spatial average with respect to $\bar{p}$ a.s., that is if
\begin{equation}{\label{kloeq1717}}
\lim_{T\rightarrow\infty}\frac{1}{T}\cdot\int_{0}^{T}f(X_{t})dt=\int_{-\infty}^{\infty}f(x)\cdot\bar{p}(x)dx
\end{equation}
for all bounded measurabe functions $f:\mathbb{R}\rightarrow\mathbb{R}$. However, (\ref{kloeq1717}) is usually quite difficult to verify directly for a diffusion process.

Recall that the Lebesgue measure space $(\mathbb{R},{\cal L},\mu_{L})$ is the completion of the Borel measure space $(\mathbb{R},{\cal B},\mu_{B})$ and ${\cal L}^{d}$ is the $d$-dimensional product space of ${\cal L}$. Let $\{{\bf X}_{t}\}_{t\geq 0}$ be a vector-valued stochastic process on a probability space $(\Omega ,{\cal F},P)$ and taking values in $\mathbb{R}^{d}$ with ${\bf X}_{t}$ being ${\cal F}\times {\cal L}^{d}$-measurable for each $t\geq 0$. Similarly, we define the transition probabilities for such $d$-dimensional process by
\begin{equation}{\label{kloeq241}}
P(s,{\bf x};t,L)=P\{{\bf X}_{t}\in L|{\bf X}_{s}={\bf x}\}
\end{equation}
for all $0\leq s<t$, ${\bf x}\in \mathbb{R}^{d}$ and $L\in {\cal L}^{d}$. Apart from the obvious change in dimension, we are now using the slightly more general Lebesgue subsets of $\mathbb{R}^{d}$
instead of the Borel subsets. The Markov property can be restated with the obvious change as the one-dimensional case. We note that the individual components of a vector Markov process need not
themselves be Markov processes. A similar feneralization applies for the definition of a vector diffusion proces. Since the transition densities need not always exist, we shall reformulate the defining properties (\ref{kloeq179})-(\ref{klieq1711}) in terms of the transition probabilities (\ref{kloeq241}) using the Lebesgue-Stieltjes integrals over subsets of $\mathbb{R}^{d}$. We require the following limits to exist for ay $\epsilon >0$, $s\geq 0$ and ${\bf x}\in \mathbb{R}^{d}$.
\begin{equation}{\label{kloeq242}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{\parallel {\bf y}-{\bf x}\parallel >\epsilon}P(s,{\bf x};t,d{\bf y})=0;
\end{equation}
\begin{equation}{\label{kloeq243}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{\parallel {\bf y}-{\bf x}\parallel\leq\epsilon}({\bf y}-{\bf x})\cdot P(s,{\bf x};t,d{\bf y})={\bf a}(s,{\bf x})
\end{equation}
and
\begin{equation}{\label{kloeq244}}
\lim_{t\downarrow s}\frac{1}{t-s}\cdot\int_{\parallel {\bf y}-{\bf x}\parallel\leq\epsilon}({\bf y}-{\bf x})\cdot ({\bf y}-{\bf x})^{T}\cdot P(s,{\bf x};t,d{\bf y})={\bf B}(s,{\bf x})\cdot {\bf B}(s,{\bf x})^{T}
\end{equation}
where ${\bf a}$ is an $d$-dimensional vector-valued functions and ${\bf D}={\bf B}{\bf B}^{T}$ is a symmetric, positive definite $d\times d$-matrix valued function. If its transition probabilities satisfy
(\ref{kloeq242})–(\ref{kloeq244}) we call the process a vector diffusion process. The {\bf drift} vector ${\bf a}$ and the diffusion matrix ${\bf D}$ have similar interpretations to their one-dimensional
counterparts, except here the off-diagonal components of ${\bf D}$ are the instantaneous rates of change in the conditioned covariances between the corresponding components of the vector process, namely
\[d_{ij}(s,{\bf x})=\lim_{t\downarrow s}\frac{1}{t-s}\cdot E\left .\left [(X_{t}^{(i)}-X_{s}^{(i)})(X_{t}^{(j))}-X_{s}^{(j)})\right |{\bf X}_{s}={\bf x}\right ].\]
When the drift vector and the diffusion matrix are moderately regular functions, the transition probabilities (\ref{kloeq241}) have densities $p(s,{\bf x};t,{\bf y})$ which satisfy the {\bf Kolmogorov forward equation}, perhaps better known as the Fokker-Planck equation,
\begin{equation}{\label{kloeq245}}
\frac{\partial p}{\partial t}+\sum_{i=1}^{d}\frac{\partial}{\partial y_{i}}(p\cdot a_{i}(t,{\bf y}))-\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}}{\partial y_{i}\partial y_{j}}(p\cdot d_{ij}(t,{\bf y}))=0
\end{equation}
($s,{\bf x}$ in $p$ fixed) with the initial condition
\[\lim_{t\downarrow s}p(s,{\bf x};t,{\bf y})=\delta ({\bf x}-{\bf y}),\]
where $\delta$ is the Dirac delta function on $\mathbb{R}^{d}$. The density $p$ is thus a fundamental solution of the parabolic partial differential equation (\ref{kloeq245}), which we can write more
compactly in operator form as
\[\frac{\partial p}{\partial t}-{\cal A}^{*}p=0\]
where ${\cal A}^{*}$ is the formal adjoint of the elliptic operator ${\cal A}$ defined as
\begin{equation}{\label{kloeq246}}
{\cal A}u(s,{\bf x})=\sum_{i=1}^{d}a_{i}(s,{\bf x})\cdot\frac{\partial u}{\partial x_{i}}(s,{\bf x})+\frac{1}{2}\sum_{i,j=1}^{d}d_{ij}(s,{\bf x})\cdot\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}(s,{\bf x}).
\end{equation}
The Kolmogorov backward equation is then
\begin{equation}{\label{kloeq247}}
\frac{\partial u}{\partial s}+{\cal A}u=0
\end{equation}
and is satisfied by $u(s,{\bf x})=p(s,{\bf x};t,{\bf y})$ for fixed $t$ and ${\bf y}$. It also has the solution
\begin{equation}{\label{kloeq248}}
u(s,{\bf x})=\mathbb{E}[f({\bf X}_{t})|{\bf X}_{s}={\bf x}]=\int_{\mathbb{R}^{d}}f({\bf y})\cdot p(s,{\bf x};t,{\bf y})d{\bf y}
\end{equation}
corresponding to the final time condition
\[\lim_{s\uparrow t}u(s,{\bf x})=f({\bf x})\]
for any sufficiently smooth function $f:\mathbb{R}^{d}\rightarrow \mathbb{R}$. Equation (\ref{kloeq248}) is often called the Kolmogorov formula.

The simplest nontrivial $d$-dimensional vector diffusion process corresponds to the zero drift vector ${\bf a}(s,{\bf x})=0$ and the identity diffusion matrix ${\bf D}(s,{\bf x})=I$. This is the $d$-dimensional vector Brownian motion ${\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(d)})$, the components of which are pairwise independent standard Brownian motions.

Strong Solutions as Diffusion Processes.

Possibly the most attractive and important property of the solutions of stochastic differential equations is that they are usually Markov processes, in fact in many cases diffusion processes. As a consequence of this we can apply the powerful analytical tools that have been developed for Markov and diffusion processes to the solutions of stochastic differential equations. In discussing such solutions below we shall always assume that we have chosen a separable version with, alomost surely, continuous sample paths. In addition, we shall often denote by $X_{t}(t_{0},x_{0})$ the solution with fixed initial value $X_{t_{0}}=x_{0}$.

Under assumptions {\bf A1}-{\bf A4}, the solution $X$ of the stochastic differential equation (\ref{kloeq451}) is a Markov process on the interval $[t_{0},T]$ with transition probabilities
\[P(s,x;t,B)=P\{X_{t}\in B|X_{s}=x\}=P\{X_{t}(s,x)\in B\}\]
for all $t_{0}\leq s\leq t\leq T$, $x\in \mathbb{R}$ and Borel subsets $B$ of $\mathbb{R}$.

For an autonomous stochastic differential equation
\[dX_{t}=a(X_{t})dt+b(X_{t})dW_{t}\]
the solutions are homogeneous Markov processes, that is their transition probabilities $p(s,x;t,B)$ depend only on the elapsed period of time $t-s\geq 0$ rather than on the specific values of $s$ and $t$. In this case, we usually write $P(x;t-s,B)$ for $P(s,x;t,B)$ and can, without loss of generality, take $s=0$. The solution of an autonomous SDE will be a stationary Markov process with stationary distribution $\tilde{P}$ if there exists a probability measure $\tilde{P}$ on $(\mathbb{R},{\cal B})$ satisfying
\[\tilde{P}(B)=\int_{\mathbb{R}}P(x;t,B)\tilde{P}(dx)\]
for all $t\geq 0$, $x\in \mathbb{R}$ and $B\in {\cal B}$. Various analytical conditions on the coefficients $a$ and $b$ are known which ensure the existence of a density function $\tilde{p}$ for such a stationary distribution as the solution of the stationary Fokker-Planck equation.

In general, the solutions of a stochastic differential equation
\begin{equation}{\label{kloeq463}}
dX_{t}=a(t,X_{t})dt+b(t,X_{t})dW_{t}
\end{equation}
are diffusion processes with their transition probabilities satisfying the three limits (\ref{kloeq179})-(\ref{kloeq1711}) for the drift $a(t,x)$ and the diffusion coefficient coefficient $b(t,x)$. We shall prove this below under assumptions {\bf A2}-{\bf A4} and the additional assumption that the coefficients of (\ref{kloeq463}) are continuous. A weakening of assumptions A2–A4 is possible provided the existence of unique solutions still holds.

Theorem. Assume that $a$ and $b$ are continuous and that A2–A4 hold. Then the solution $X_{t}$ of $(\ref{kloeq463})$ for any fixed initial value $X_{t_{0}}$ is a diffusion process on $[t_{0},T]$ with drift $a(t,x)$ and diffusion coefficient $b(t,x)$. $\sharp$

Diffusion Processes as Weak Solutions.

We shall now turn to the converse problem of whether or not a given diffusion process with probability density satisfying the Fokker-Planck equation is the solution of some stochastic differential equation. Closely
related to this is the question of the existence of weak solutions of a stochastic differential equation. We recall that a weak solution of a stochastic differential equation for which the coefficients of the equation, but not the Brownian motions, are specified. For a diffusion process these coefficients are obtained from the drift $a(t,x)$ and diffusion term $\sigma (t,x)=b^{2}(t,x)$ of the process. On a superficial level, the
question has a simple affirmative answer: if $Y_{t}$ is the diffusion process with drift $a$ and diffusion term $\sigma =b^{2}$, or even if just these coefficients are specified, then we can take the stochastic
differential equation
\begin{equation}{\label{kloeq471}}
dX_{t}=a(t,X_{t})dt+b(t,X_{t})dW_{t}
\end{equation}
with the initial value $X_{t_{0}}=Y_{t_{0}}$ for any Brownian motion $\{W_{t}\}_{t\geq 0}$. Provided the coefficients satisfy appropriate properties, such as assumptions A1–A3 of Theorem \ref{klot453},
there will be a solution $X$ for each Brownian motion. Such a solution is then an equivalent stochastic process to the given diffusion process $Y$, that is it has the same probability law. In general, however, it will not be sample path equivalent to $Y_{t}$. To guarantee this we must choose the Brownian motion with more care.

Let $Y$ be a given diffusion proces on $[0,T]$ with drift $a(t,y)$ and strictly positive diffusion coefficient $b(t,y)$. Under assumptions we shall specify later we define functions $g$ and $\bar{a}$ by
\begin{equation}{\label{kloeq472}}
g(t,y)=\int_{0}^{y}\frac{1}{b(t,x)}dx\mbox{ and }\bar{a}(t,z)=\left (\frac{\partial g}{\partial t}+a\cdot\frac{\partial g}{\partial y}+\frac{b^{2}}{2}\cdot\frac{\partial^{2}g}{\partial y^{2}}\right )(t,g^{-1}(t,z))
\end{equation}
with $a$ and $b$ evaluated at $(t,y)$, where $y=g^{-1}(t,z)$ is the inverse of $z=g(t,y)$. Then we define a process $Z_{t}=g(t,Y_{t})$, which is a diffusion process with drift $\bar{a}(t,z)$ and diffusion coefficient $1$, and a process
\begin{equation}{\label{kloeq474}}
\widetilde{W}_{t}=Z_{t}-Z_{0}-\int_{0}^{t}\bar{a}(s,Z_{s})ds,
\end{equation}
which will turn out to be a Brownian motion. Consequently (\ref{kloeq474}) will be equivalent to the stochastic differential equation
\[dZ_{t}=\bar{a}(t,Z_{t})dt+1d\widetilde{W}_{t},\]
which, by (\ref{kloeq472}) and Ito’s formula, will imply that $Y_{t}$ is a solution of the stochastic differential equation
\begin{equation}{\label{kloeq475}}
dY_{t}=a(t,Y_{t})dt+b(t,Y_{t})d\widetilde{W}_{t},
\end{equation}
that is of (\ref{kloeq471}) with the Brownian motion $\widetilde{W}_{t}$.

Theorem. (Kloeden and Platen \cite{klo}). Let $Y$ be a diffusion process on $[0,T]$ with coefficients $a(t,y)$ and $b(t,y)>0$ satisfying for all $y\in \mathbb{R}$ and $t\in [0,T]$, the following conditions

$a(t,y)$ is continuous in both variables and $|a(t,y)|\leq K\cdot (1+|y|)$ for some positive constants $K$;
$b(t,y)$ is continuous in both variables, $b^{-1}(t,y)$ is bounded, and the partial derivatives $\partial b/\partial t$ and $\partial b/\partial y$ are continuous and bounded;
There exists a function $\psi (y)>1+|y|$ such that
\begin{align*}
&\sup_{0\leq t\leq T} E[\psi (Y_{t})]<\infty\\
& \mathbb{E}[|Y_{t}-Y_{s}|Y_{s}=y]+\mathbb{E}[|Y_{t}-Y_{s}|^{2}|Y_{s}=y]\leq (t-s)\cdot\psi (y)\\
& \mathbb{E}[|Y_{t}||Y_{s}=y]+\mathbb{E}[|Y_{t}|^{2}|Y_{s}=y]\leq\psi (y).
\end{align*}

Then $\widetilde{W}$ defined by $(\ref{kloeq474})$ is a Brownian motion and $Y$ is a solution of the stochastic differential equation (\ref{kloeq475}). $\sharp$

The rather strong conditions in the preceding theorem are sufficient, but nor necessary for the existence of weak solutions. We remark that in some cases the SDE have only weak solutions and no strong solution.

Example. The stochastic differential equation
\[dX_{t}=sign(X_{t})dt+dW_{t}\]
where $sign(x)=1$ if $x\geq 0$ and $-1$ if $x<0$, only has weak solutions, but no strong solution for the initial value $X_{0}=0$. In fact, if $X_{t}$ is such a weak solution for the Brownian motion $W$, then $-X_{t}$ is a weak solution for the Brownian motion $-W$. These solutions have the same probability law, but not the same sample paths. $\sharp$

Vector Stochastic Differential Equations.

We consider an $m$-dimensional Brownian motion $\{W_{t}\}_{t\geq 0}$ with components $W_{t}^{(1)},\cdots ,W_{t}^{(m)}$, which are independent scalar Brownian motions with respect to a common family of $\sigma$-fields $\{{\cal F}_{t}\}_{t\geq 0}$. Then we take a $d$-dimensional function ${\bf a}:[0,T]\times\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ and a $d\times m$-matrix function ${\bf b}:[0,T]\times \mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times m}$ to form a $d$-dimensional vector stochastic differential equation
\begin{equation}{\label{kloeq481}}
d{\bf X}_{t}={\bf a}(t,{\bf X}_{t})+{\bf b}(t,{\bf X}_{t})d{\bf W}_{t}.
\end{equation}
We interpret this as a stochastic integral equation
\begin{equation}{\label{kloeq482}}
{\bf X}_{t}={\bf X}_{t_{0}}+\int_{t_{0}}^{t}{\bf a}(s,{\bf X}_{s})ds+\int_{t_{0}}^{t}{\bf b}(s,{\bf X}_{s})d{\bf W}_{s}
\end{equation}
where the Lebesgue and Ito integrals are determined component by component, with the $i$th component of (\ref{kloeq482}) being
\[X_{t}^{(i)}=X_{t_{0}}^{(i)}+\int_{t_{0}}^{t}a_{i}(s,{\bf X}_{s})ds+\sum_{j=1}^{m}\int_{t_{0}}^{t}b_{ij}(s,{\bf X}_{s})dW_{s}^{(j)}.\]
Analogous definition of strong and weak solutions apply here with the resulting process ${\bf X}_{t}$ required to be ${\cal F}_{t}$-measurable. The existence and uniqueness theorem for strong solutions,
Theorem~\ref{klot453} carries over verbatim to the vector case provided the absolute values in the assumptions and proof are replaced by vector and matrix norms.

When the coefficient of a scalar equation also depend explicitly on $W_{t}$ as in
\[dY_{t}=a(t,Y_{t},W_{t})dt+b(t,Y_{t},W_{t})dW_{t},\]
we can rewrite the equation as the $2$-dimensional vector SDE
\[d{\bf X}_{t}=\left [\begin{array}{c}a(t,{\bf X}_{t})\\ 0\end{array}\right ]dt+\left [\begin{array}{c}b(t,{\bf X}_{t})\\ 1\end{array}\right ]d{\bf W}_{t}\]
with state components $X_{t}^{(1)}=Y_{t}$ and $X_{t}^{(2)}=W_{t}$. Another situation in which vector stochastic differential equations arise is when processes are restricted to lie on certain manifolds, such as
the unit circle $S^{1}$ which is a one-dimensional compact manifold. A Brownian motion on $S^{1}$ satisfies the vector differential equation
\[d{\bf X}_{t}=-\frac{1}{2}X_{t}dt+\left [\begin{array}{cc}
0 & -1\\ 1 & 0\end{array}\right ]\cdot {\bf X}_{t}dW_{t},\]
with the constraint $\parallel {\bf X}_{t}\parallel =1$, where $W_{t}$ is a scalar Brownian motion.

For a sufficiently smooth transformation ${\bf U}:[0,T]\times\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ of the solution ${\bf X}_{t}$ of (\ref{kloeq481}) we obtain a $k$-dimensional process
${\bf Y}_{t}={\bf U}(t,{\bf X}_{t})$. This process will have a vector stochastic differential which can be determined by applying the Ito formula to each component. The resulting expression is more
transparent in component form
\[dY_{t}^{(p)}=\left (\frac{\partial U_{p}}{\partial t}+\sum_{i=1}^{d}a_{i}\cdot\frac{{\partial U}_{p}}{\partial x_{i}}+\frac{1}{2}\sum_{i,j=1}^{d}\sum_{l=1}^{m}b_{il}b_{jl}\cdot\frac{\partial^{2}U_{p}}
{\partial x_{i}\partial x_{j}}\right )dt+\sum_{l=1}^{m}\sum_{i=1}^{d}b_{il}\cdot\frac{{\partial U}_{p}}{\partial x_{i}}dW_{t}^{(l)}\]
for $p=1,2,\cdots ,k$ where the terms are all evaluated at $(t,{\bf X}_{t})$. As in the scalar case, we can use this formula to determine the solution of certain vector stochastic differential equations in terms of known solutions of other equations, for example linear equations.

The general form of a $d$-dimensional linear stochastic differential equation is
\begin{equation}{\label{kloeq484}}
d{\bf X}_{t}=({\bf A}_{t}+{\bf a}_{t})dt+\sum_{l=1}^{m}({\bf B}_{t}^{(1)}\cdot {\bf X}_{t}+{\bf b}^{(l)}_{t})dW^{(l)}_{t}
\end{equation}
where ${\bf A}_{t},{\bf B}_{t}^{(1)},{\bf B}_{t}^{(2)},\cdots , {\bf B}_{t}^{(m)}$ are $d\times d$-matrix functions and ${\bf a}_{t}, {\bf b}^{(1)}_{t},{\bf b}^{(2)}_{t},\cdots ,{\bf b}^{(m)}_{t}$ are $d$-dimensional vector functions. When the ${\bf B}^{(l)}$ are all identically zero, we say that (\ref{kloeq484}) is linear in the narrow-sense and when ${\bf a}$ and the ${\bf b}^{(l)}$ are all zero we call it a homogeneous equation. Duplicating the argument used for the scalar case, we find that the solution of (\ref{kloeq484}) is
\begin{equation}{\label{kloeq485}}
{\bf X}_{t}=\Phi_{t,t_{0}}\cdot\left ({\bf X}_{t_{0}}+\int_{t_{0}}^{t}\Phi^{-1}_{s,t_{0}}\cdot\left ({\bf a}(s)-\sum_{l=1}^{m}{\bf B}_{s}^{(l)}\cdot {\bf b}_{s}^{(l)}\right )ds+\sum_{l=1}^{m}\int_{t_{0}}^{t}\Phi_{s,t_{0}}^{-1}\cdot {\bf b}_{s}^{(l)}dW_{s}^{(l)}\right ),
\end{equation}
where $\Phi_{t,t_{0}}$ is the $d\times d$ fundamental matrix satisfying $\Phi_{t_{0},t_{0}}=I$ and the homogeneous matrix stochastic differential equation
\[d\Phi_{t,t_{0}}={\bf A}_{t}\cdot\Phi_{t,t_{0}}dt+\sum_{l=1}^{m}{\bf B}_{t}^{(l)}\cdot\Phi_{t,t_{0}}dW_{t}^{(l)}.\]
Unlike the scalar homogeneous linear equations, we cannot generally solve (\ref{kloeq485}) explicitly for the fundamental solution, even when all of the matrices are constant matrices. If, however, the matrices
${\bf A},{\bf B}^{(1)},\cdots ,{\bf B}^{(m)}$ are constants and commute, that is, if
\[{\bf A}{\bf B}^{(l)}={\bf B}^{(l)}{\bf A}\mbox{ and }{\bf B}^{(l)}{\bf B}^{(k)}={\bf B}^{(k)}{\bf B}^{(l)}\]
for all $k=1,2,\cdots ,m$, then we obtain the following explicit expression for the fundamental matrix solution
\[\Phi_{t,t_{0}}=\exp\left (\left ({\bf A}-\frac{1}{2}\sum_{l=1}^{m}({\bf B}^{(l)})^{2}\right )\cdot (t-t_{0})+\sum_{l=1}^{m}{\bf B}^{(l)}\cdot (W_{t}^{(l)}-W_{t_{0}}^{(l)})\right ).\]
In the spacial case that (\ref{kloeq484}) is linear in the narrow-sense this reduces to
\[\Phi_{t,t_{0}}=\exp\left ({\bf A}\cdot (t-t_{0})\right ),\]
which is the funcdamental matrix of the deterministic linear system $\dot{{\bf x}}={\bf Ax}$. In such autonomous cases $\Phi_{t,t_{0}}=\Phi_{t-t_{0},0}$, so we need only to consider $t_{0}=0$ and can write
$\Phi_{t}$ for $\Phi_{t,0}$. In the same way as for scalar linear SDE, we can derive vector and matrix ordinary differential equations for the vector mean ${\bf m}(t)=\mathbb{E}[{\bf X}_{t}]$ and the $d\times d$ matrix second moment ${\bf P}(t)=\mathbb{E}[{\bf X}_{t}{\bf X}_{t}^{T}]$ of a general vector linear SDE (\ref{kloeq484}). Thus ${\bf P}_{t}$ is a symmetric matrix. We then obtain
\[\frac{d{\bf m}}{dt}={\bf A}_{t}\cdot {\bf m}+{\bf a}_{t}\]
and
\begin{align*}
\frac{d{\bf P}}{dt} & ={\bf A}_{t}\cdot {\bf P}+{\bf P}\cdot {\bf A}_{t}^{T}+\sum_{l=1}^{m}{\bf B}_{t}^{(l)}\cdot {\bf P}\cdot ({\bf B}_{t}^{(l)})^{T}+{\bf a}_{t}\cdot ({\bf m}(t))^{T}+{\bf m}(t)\cdot {\bf a}_{t}^{T}\\
& +\sum_{l=1}^{m}\left ({\bf B}_{t}^{(l)}\cdot {\bf m}(t)\cdot ({\bf b}_{t}^{(l)})^{T}+{\bf b}_{t}^{(l)}\cdot ({\bf m}(t))^{T}\cdot {\bf B}_{t}^{(l)}+{\bf b}_{t}^{(l)}\cdot ({\bf b}_{t}^{(l)})^{T}\right ),
\end{align*}
with initial conditions ${\bf m}(t_{0})=E[{\bf X}_{t_{0}}]$ and ${\bf P}(t_{0})=E[{\bf X}_{t_{0}}\cdot {\bf X}_{t_{0}}^{T}]$.

The solution ${\bf X}_{t}$ of a vector stochastic differential equation (\ref{kloeq481}) is a Markov process. Under smoothness conditions on its coefficients, like those in the existence and uniqueness Theorem \ref{klot453}, ${\bf X}_{t}$ is also a diffusion process with drift vector ${\bf a}(t,{\bf x})$ and $d\times d$ diffusion matrix ${\bf D}(t,{\bf x})={\bf b}(t,{\bf x})\cdot ({\bf b}(t,{\bf x}))^{T}$, or in
component form
\[d_{ij}(t,{\bf x})=\sum_{l=1}^{m}b_{il}(t,{\bf x})\cdot {\bf b}_{jl}(t,{\bf x})\mbox{ for }i,j=1,2,\cdots ,d.\]
The transition probabilities then satisfy the Kolmogorov backward equation with these coefficients. Generally, the diffusion matrices ${\bf D}$ are only positive semidefinite rather than positive definite, and this may lead to singularities in the transition densities. Many difficulties can thus arise in functional analytical investigations of these partial differential equations, but, as we shall see, these can often be circumvented by the use of probabilistic methods. An important situation where this happens is in the verification of that the Kolmogorov formula
\begin{equation}{\label{kloeq4811}}
u(s,{\bf x})=E[f({\bf X}_{T})|{\bf X})_{s}={\bf x}]
\end{equation}
gives a solution of the Kolmogorov backward equation (\ref{kloeq247})
\begin{equation}{\label{kloeq4812}}
\frac{\partial u}{\partial s}+{\cal A}u=0
\end{equation}
for $0\leq s\leq T$, where ${\cal A}$ is the elliptic operator (\ref{kloeq246}) with the final condition
\begin{equation}{\label{kloeq4813}}
u(T,{\bf x})=f({\bf x})
\end{equation}
for a sufficiently smooth function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$.

To be more specific, let $C^{l}(\mathbb{R}^{d},\mathbb{R})$ denote the space of $l$ times continuously differentiable functions $w:\mathbb{R}^{d}\rightarrow \mathbb{R}$ and $C_{P}^{l}(\mathbb{R}^{d},\mathbb{R})$ the subspace of functions $w\in C^{l}(\mathbb{R}^{d},\mathbb{R})$ for which all partial derivatives up to order $l$ have polynomial growth, that is for which there exist constants $K>0$ and $r\in\{1,2,3,\cdots\}$, depending on $w$, such that
\[|\partial^{i}_{{\bf y}}w({\bf y})|\leq K\cdot (1+\parallel {\bf y}\parallel^{2r})\]
for all ${\bf y}\in \mathbb{R}^{d}$ and any partial derivative $\partial^{i}_{{\bf y}}w$ of order $j\leq l$.

Theorem. (Kloeden and Platen \cite{klo}) Suppose that $f\in C_{P}^{2(r+1)}(\mathbb{R}^{d},\mathbb{R})$ for some $r=1,2,\cdots$ and that ${\bf X}_{t}$ is a homogeneous diffusion process for which
the drift vector and diffusion matrix components $a_{i},b_{ij}\in C_{P}^{2(r+1)}(\mathbb{R}^{d},\mathbb{R})$ with uniformly bounded derivatives. Then $latex u:[0,T]\times \mathbb{R}^{d}\rightarrow
\mathbb{R}$ defined by $(\ref{kloeq4811})$ satisfies the final value problem $(\ref{kloeq4812})$ and $(\ref{kloeq4813})$ with $\partial u/\partial s$ continuous and $u(s,\cdot )\in C_{P}^{2(r+1)}(\mathbb{R}^{d},\mathbb{R})$ for each $0\leq s\leq T$. $\sharp$

We remark that the diffusion process may be degenerate here, that is with the diffusion matrix vanishing at various points in $\mathbb{R}^{d}$. The Ito formula is used to show that the function $u$ given by (\ref{kloeq4811}) is a solution of the Kolmogorov backward equation (\ref{kloeq4812}). It can be used in a similar way under analogous smoothness assumptions to show that the Feynman-Kac formula
\[u(s,{\bf x})=E\left .\left [f({\bf X}_{T})\cdot\exp\left (\int_{s}^{T}g({\bf X}_{u})du\right )\right |{\bf X}_{s}={\bf x}\right ],\]
where $g$ is a bounded function, is a solution of the partial differential equation
\[\frac{\partial u}{\partial s}+{\cal A}u+gu=0\]
with the final condition (\ref{kloeq4813}) and elliptic opeartor (\ref{kloeq246}).

In nonlinear filtering and other applications we often encounter a “drifted” Brownian motion ${\bf X}$ on a probability space $(\Omega ,{\cal F},P)$ with a filtration $\{{\cal F}_{t}\}_{0\leq t\leq T}$. This is defined by
\[{\bf X}_{t}={\bf X}_{0}+\int_{0}^{t}{\bf A}_{s}ds+{\bf W}_{t}\]
for $t]in [0,T]$, where ${\bf A}_{s}$ is an ${\cal F}_{s}$-adapted, right-continuous process and ${\bf W}_{s}$ is an ${\cal F}_{s}$-adapted Brownian motion with respect to the probability measure $\mathbb{P}$. In general, ${\bf X}$ is not a Brownian motion with respect to the given probability measure, but it is sometimes useful to interpret it as one with respect to another probability measure. We can do this by using the Girsanov transformation to transform the underlying probability measure $P$ on the canonical sample space $\Omega =C_{0}([0,T],\mathbb{R})$ to an absolutely continuous probability measure $\mathbb{P}_{X}$ with the Radon-Nikodym derivative
\[\frac{dP_{X}}{dP}=\exp\left (\int_{0}^{T}{\bf A}_{s}d{\bf X}_{s}-\frac{1}{2}\int_{0}^{T}\parallel {\bf A}_{s}\parallel^{2}ds\right ).\]
The ${\cal F}_{s}$-adapted process ${\bf X}_{s}$ turns out then to be a Brownian motion on the canonical probability space $(\Omega ,{\cal F},\mathbb{P}_{X})$.

We say that a $d$-dimensional Ito process $\{{\bf X}_{t}\}_{t\geq 0}$ is ergodic if it has a unique invariant probability law $\mu$ such that
\[\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}f({\bf X}_{s}^{0,{\bf x}})ds=\int_{\mathbb{R}^{d}}f({\bf y})d\mu ({\bf y})\]

a.s. for any $\mu$-integrable function $f:\mathbb{R}^{d}\rightarrow \mathbb{R}$ and any deterministic initial condition ${\bf X}_{0}={\bf x}$.

Theorem. (Kloeden and Platen \cite{klo}) Suppose that the drift ${\bf a}$ and diffusion coefficient ${\bf b}$ of an autonomous Ito process ${\bf X}$ are smooth with bounded derivatives of any order, $latex {\bf
b}$ is bounded and there exists a constant $\beta >0$ and a compact subset $K\subset \mathbb{R}^{d}$ such that
\[{\bf x}^{T}\cdot {\bf a}({\bf x})\leq -\beta\cdot\parallel {\bf x}\parallel^{2}\]
for all ${\bf x}\in \mathbb{R}^{d}\setminus K$. Then ${\bf X}$ is ergodic. $\sharp$

The Markov Property.

In a stochastic differential equation of the form
\[d{\bf X}_{t}={\bf b}(t,{\bf X}_{t})dt+\boldsymbol{\sigma}(t,{\bf X}_{t})d{\bf W}_{t},\]

where ${\bf X}_{t}\in \mathbb{R}^{n}$, ${\bf b}(t,{\bf x})\in \mathbb{R}^{n}$, $\boldsymbol{\sigma}(t,{\bf x})\in \mathbb{R}^{n\times m}$ and ${\bf W}_{t}$ is $m$-dimensional Brownian motion, we will call ${\bf b}$ the drift coefficient and $\boldsymbol{\sigma}$, or sometimes $\frac{1}{2}\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}$, the diffusion coefficient. Thus the solution of a stochastic differential equation may be thought of as the mathematical description of the motion of a small particle in a moving fluid. Therefore such stochastic processes are called (Ito)
diffusions.

Definition. A $($time-homogeneous$)$ Ito diffusion is a stochastic process ${\bf X}_{t}(\omega ):[0,\infty )\times\Omega\rightarrow \mathbb{R}^{n}$ satisfying a stochastic differential equation of the form
\begin{equation}{\label{okseq714}}
d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}\mbox{ for }t\geq s, {\bf X}_{s}={\bf x}
\end{equation}
where ${\bf W}_{t}$ is $m$-dimensional Brownian motion and ${\bf b}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $\boldsymbol{\sigma}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times m}$ satisfy the conditions in Theorem~\ref{okst521}, which in this case simplify to
\[\parallel {\bf b}({\bf x})-{\bf b}({\bf y})\parallel +\parallel\boldsymbol{\sigma}({\bf x})-\boldsymbol{\sigma}({\bf y})\parallel\leq d\cdot\parallel {\bf x}-{\bf y}\parallel\mbox{ for }{\bf x},{\bf y}\in \mathbb{R}^{n}\]
where $\parallel\boldsymbol{\sigma}\parallel =\sum |\sigma_{ij}|^{2}$. $\sharp$

We will denote the (unique) solution of (\ref{okseq714}) by ${\bf X}_{t}={\bf X}_{t}^{s,{\bf x}}$ for $t\geq s$. If $s=0$ we write ${\bf X}_{t}^{\bf x}$ for ${\bf X}_{t}^{0,{\bf x}}$. Note that we have assumed in (\ref{okseq714}) that ${\bf b}$ and $\boldsymbol{\sigma}$ do not depend on $t$ but on ${\bf x}$ only. We shall see later that the general case can be reduced to this situation. The resulting process
${\bf X}_{t}$ will have the property of being {\bf time-homogeneous}, in the following sense:

Note that
\[{\bf X}_{s+h}^{s,{\bf x}}={\bf x}+\int_{s}^{s+h}{\bf b}({\bf X}_{u}^{s,{\bf x}})du+\int_{s}^{s+h}\boldsymbol{\sigma}({\bf X}_{u}^{s,{\bf x}})d{\bf W}_{u}={\bf x}+\int_{0}^{h}{\bf b}
({\bf X}_{s+v}^{s,{\bf x}})dv+\int_{0}^{h}\boldsymbol{\sigma}({\bf X}_{s+v}^{s,{\bf x}})d\widetilde{{\bf W}}_{v},\]
($u=s+v$), where $\widetilde{{\bf W}}_{v}={\bf W}_{s+v}-{\bf W}_{v}$ for $v\geq 0$. On the other hand of course
\[{\bf X}_{h}^{\bf x}={\bf x}+\int_{0}^{h}{\bf b}({\bf X}_{v}^{\bf x})dv+\int_{0}^{h}\boldsymbol{\sigma}({\bf X}_{v}^{\bf x})d{\bf W}_{v}.\]
Since $\{\widetilde{{\bf W}}_{v}\}_{v\geq 0}$ and $\{{\bf W}_{v}\}_{v\geq 0}$ have the same $P^{0}$-distribution, it follows by weak uniqueness in Proposition \ref{oksl531} of the solution of the stochastic differential equation
\[d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}\mbox{ with }{\bf X}_{0}={\bf x}\]
that $\{{\bf X}_{s+h}^{s,{\bf x}}\}_{h\geq 0}$ and $\{{\bf X}_{h}^{\bf x}\}_{h\geq 0}$ have the same $P^{0}$-distribution, i.e., $\{{\bf X}_{t}\}_{t\geq 0}$ is time-homogeneous.

We now introduce the probability laws $Q^{\bf x}$ of $\{{\bf X}_{t}\}_{t\geq 0}$ for ${\bf x}\in \mathbb{R}^{n}$. Intuitively, $Q^{\bf x}$ gives the distribution of $\{{\bf X}_{t}\}_{t\geq 0}$ assuming that ${\bf X}_{0}={\bf x}$. To express this mathematically, we let ${\cal M}_{\infty}$ be the $\sigma$-filed generated by the random variables $\omega\mapsto {\bf X}_{t}(\omega )={\bf X}_{t}^{\bf y}(\omega )$, where $t\geq 0$ and ${\bf y}\in\mathbb{R}^{n}$. Define $Q^{\bf x}$ on the members of ${\cal M}$ by
\[Q^{\bf x}\{{\bf X}_{t_{1}}\in E_{1},\cdots ,{\bf X}_{t_{k}}\in E_{k}\}=P^{0}\{{\bf X}_{t_{1}}^{\bf x}\in E_{1},\cdots ,{\bf X}_{t_{k}}^{\bf x}\in E_{k}\}\]

where $E_{i}\subset \mathbb{R}^{n}$ are Borel sets for $1\leq i\leq k$. As before we let ${\cal F}_{t}^{(m)}$ be the $\sigma$-field generated by $\{{\bf W}_{s}:s\leq t\}$. Similarly we let ${\cal M}_{t}$ be the $\sigma$-field generated by $\{{\bf X}_{s}:s\leq t\}$. We have established earlier (see Theorem \ref{okst521}) that ${\bf X}_{t}$ is measurable with respect to ${\cal F}_{t}^{(m)}$, so ${\cal M}_{t}\subseteq {\cal F}_{t}^{(m)}$. We now prove that ${\bf X}_{t}$ satisfies the imporatnt {\bf Markov property}: The future behavior of the process given what has happened up to time $t$ is the same as the behavior obtained when starting the process at ${\bf X}_{t}$.

\begin{equation}{\label{okst712}}\tag{5}\mbox{}\end{equation}

Theorem \ref{okst712}. (Oksendal \cite{oks} The Markov property for Ito diffusions) Let $f$ be a bounded Borel function from $\mathbb{R}^{n}$ to $\mathbb{R}$. Then, for $t,h\geq 0$, we have
\begin{equation}{\label{okseq718}}
\mathbb{E}^{\bf x}\left .\left [f({\bf X}_{t+h})\right |{\cal F}_{t}^{(m)}\right ](\omega )=\mathbb{E}^{{\bf X}_{t}(\omega )}[f({\bf X}_{h})]. \sharp
\end{equation}

Here and in the following $E^{\bf x}$ denotes the expectation with respect to the probability measure $Q^{\bf x}$. Thus $E^{\bf y}[f({\bf X}_{h})]$ means $E[f({\bf X}_{h}^{\bf y})]$, where $E$ denotes the expectation with respect to the measure $P^{0}$. The right hand side means the function $\mathbb{E}^{\bf y}[f({\bf X}_{h})]$ evaluated at ${\bf y}={\bf X}_{t}(\omega )$.

Theorem \ref{okst712} states that ${\bf X}_{t}$ is a Markov process with respect to the family of $\sigma$-fields $\{{\cal F}_{t}^{(m)}\}_{t\geq 0}$. Note that since ${\cal M}_{t}\subseteq {\cal F}_{t}^{(m)}$ this implies that ${\bf X}_{t}$ is also a Markov process with respect to the $\sigma$-fileds $\{{\cal M}_{t}\}_{t\geq 0}$. This follows from
\[\mathbb{E}^{\bf x}[f({\bf X}_{t+h})|{\cal M}_{t}]=\mathbb{E}^{\bf x}[\mathbb{E}^{\bf x}[f({\bf X}_{t+h})|{\cal F}_{t}^{(m)}]|{\cal M}_{t}]=\mathbb{E}^{\bf x}[\mathbb{E}^{{\bf X}_{t}}[f({\bf X}_{h})]|{\cal M}_{t}]=\mathbb{E}^{{\bf X}_{t}}[f({\bf X}_{h})]\]
since $\mathbb{E}^{{\bf X}_{t}}[f({\bf X}_{h})]$ is ${\cal M}_{t}$-measurable.

The Strong Markov Property.

Roughly, the strong Markov property states that relation of the form (\ref{okseq718}) consitutes to hold if the time $t$ is replaced by a random time $T (\omega )$ of a more general type called stopping time. If $H\subseteq \mathbb{R}^{n}$ is any set we define the first exit time from $H$, $T_{H}$, as follows
\[T_{H}=\inf\{t>0:{\bf X}_{t}\not\in H\}.\]
If we include the sets of measure $0$ in ${\cal M}_{t}$ (which we do) then the family $\{{\cal M}_{t}\}$ is right-continuous, i.e., ${\cal M}_{t}={\cal M}_{t+}$, where ${\cal M}_{t+}=\bigcap_{s>t}{\cal M}_{t}$.

Let $T$ be a stopping time with respect to $\{{\cal N}_{t}\}_{t\geq 0}$ and let ${\cal N}_{\infty}$ be the smallest $\sigma$-field containing ${\cal N}_{t}$ for all $t\geq 0$. Then the $\sigma$-field ${\cal N}_{T}$ consists of all sets $N\in {\cal N}_{\infty}$ such that $N\bigcap\{T\leq t\}\in {\cal N}_{t}$ for all $t\geq 0$. In the case when ${\cal N}_{t}={\cal M}_{t}$, an alternative and more intuitive description is
\[{\cal M}_{T}=\mbox{the $\sigma$-field generated by $\{X_{s\wedge T}:s\geq 0\}$}.\]
Similarly, if ${\cal N}_{t}={\cal F}_{t}^{(m)}$, we get
\[\mbox{${\cal F}_{t}^{(m)}=$ the $\sigma$-field generated by $\{W_{s\wedge T}:s\geq 0\}$}\]

\begin{equation}{\label{okst724}}\tag{6}\mbox{}\end{equation}

Theorem \ref{okst724}. (Oksendal \cite{oks})(The strong Markov property for Ito diffusions) Let $f$ be a bounded Borel function on $\mathbb{R}^{n}$, $T$ a stopping time with respect to $latex {\cal
F}_{t}^{(m)}$ for $T<\infty$ a.s. Then
\begin{equation}{\label{okseq722}}
\mathbb{E}^{\bf x}[f({\bf X}_{T+h})|{\cal F}_{T}^{(m)}]=\mathbb{E}^{{\bf X}_{T}}[f({\bf X}_{h})]
\end{equation}
for all $h\geq 0$. $\sharp$

We now extend (\ref{okseq722}) to the following: If $f_{1},\cdots ,f_{k}$ are bounded Borel functions on $\mathbb{R}^{n}$ and $T$ is an ${\cal F}_{t}^{(m)}$-stopping time for $T<\infty$ a.s. then
\[\mathbb{E}^{\bf x}[f_{1}({\bf X}_{T+h_{1}})f_{2}({\bf X}_{T+h_{2}}\cdots f_{k}({\bf X}_{T+h_{k}})|{\cal F}_{T}^{(m)}]=\mathbb{E}^{{\bf X}_{T}}[f_{1}({\bf X}_{h_{1}}\cdots f_{k}({\bf X}_{h_{k}}]\]
for all $0\leq h_{1}\leq h_{2}\leq\cdots\leq h_{k}$. This follows by induction. To illustrate the argument we prove it in the case $k=2$
\begin{align*}
\mathbb{E}^{\bf x}[f_{1}({\bf X}_{T+h_{1}})f_{2}({\bf X}_{T+h_{2}})| {\cal F}_{T}^{(m)}] & = \mathbb{E}^{\bf x}[\mathbb{E}^{\bf x}[f_{1}({\bf X}_{T+h_{1}})f_{2}({\bf X}_{T+h_{2}})|
{\cal F}_{T+h_{1}}]|{\cal F}_{T}^{(m)}]\\
& = \mathbb{E}^{\bf x}[f_{1}({\bf X}_{T+h_{1}})\cdot \mathbb{E}^{\bf x}[f_{2}({\bf X}_{T+h_{2}})|{\cal F}_{T+h_{1}}]|{\cal F}_{T}^{(m)}]\\
& = \mathbb{E}^{\bf x}[f_{1}({\bf X}_{T+h_{1}})\cdot \mathbb{E}^{{\bf X}_{T+h_{1}}}[f_{2}({\bf X}_{h_{2}-h_{1}}]|{\cal F}_{T}^{(m)}]\\
& =\mathbb{E}^{{\bf X}_{T}}[f_{1}({\bf X}_{h_{1}})\cdot \mathbb{E}^{{\bf X}_{h_{1}}}[f_{2}({\bf X}_{h_{2}-h_{1}}]]\\
& =\mathbb{E}^{{\bf X}_{T}}[f_{1}({\bf X}_{h_{1}})\cdot \mathbb{E}^{\bf x}[f_{2}({\bf X}_{h_{2}}|{\cal F}_{h_{1}}^{(m)}]]\\
& =\mathbb{E}^{{\bf X}_{T}}[f_{1}({\bf X}_{h_{1}})f_{2}({\bf X}_{h_{2}})].
\end{align*}

The Generator of an Ito Diffusion.

It is fundamental for many applications that we can associate a second order partial differential operator $A$ to an Ito diffusion ${\bf X}_{t}$. The basic connection between $A$ and ${\bf X}_{t}$ is that $A$ is the generator of the process ${\bf X}_{t}$.

Definition. Let $\{{\bf X}_{t}\}$ be a (time-homogeneous) Ito diffusion in $\mathbb{R}^{n}$. The $($infinitesimal$)$ generator $A$ of ${\bf X}_{t}$ is defined by
\[Af({\bf x})=\lim_{t\downarrow 0}\frac{E^{\bf x}[f({\bf X}_{t})]-f({\bf x})}{t}\mbox{ for }{\bf x}\in \mathbb{R}^{n}.\]

The set of functions $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ such that the limit exists at ${\bf x}$ is denoted by ${\cal D}_{A}({\bf x})$, while ${\cal D}_{A}$ denotes the set of functions for which the
limit exists for all ${\bf x}\in \mathbb{R}^{n}$. $\sharp$

To find the relation between $A$ and the coefficients ${\bf b}$ and $\boldsymbol{\sigma}$ in the stochastic differential equation (\ref{okseq714}) defining ${\bf X}_{t}$ we need the following result, which is useful in many connections.

Proposition. (Oksendal \cite{oks}) Let ${\bf Y}_{t}={\bf Y}_{t}^{\bf x}$ be an Ito process in $\mathbb{R}^{n}$ of the form
\[{\bf Y}_{t}^{\bf x}(\omega )={\bf x}+\int_{0}^{t}{\bf u}(s,\omega )ds+\int_{0}^{t}{\bf v}(s,\omega )d{\bf W}_{s}(\omega )\]

where ${\bf W}$ is $m$-dimensional Brownian. Let $f\in C_{0}^{2}(\mathbb{R}^{n})$ and $T$ be a stopping time with respect to $\{{\cal F}_{t}^{(m)}\}_{t\geq 0}$, and assume that $latex E^{\bf
x}[T]<\infty$. Assume that ${\bf u}(t,\omega )$ and ${\bf v}(t,\omega )$ are bounded on the set of $(t,\omega )$ such that ${\bf Y}_{t}(\omega )$ belongs to the support of $f$. Then
\[\mathbb{E}^{\bf x}[f({\bf Y}_{T})]=f({\bf x})+\mathbb{E}^{\bf x}\left [\int_{0}^{T}\left (\sum_{i}{\bf u}(s,\omega )\cdot\frac{\partial f}{\partial x_{i}}({\bf Y}_{s})+
\frac{1}{2}\sum_{i,j}({\bf v}{\bf v}^{T})_{ij}(s,\omega )\cdot\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf Y}_{s})\right )ds\right ],\]
where $\mathbb{E}^{\bf x}$ is the expectation with respect to the natural probability law $\mathbb{P}^{\bf x}$ for ${\bf Y}_{t}$ starting at ${\bf x}$:
\[\mathbb{P}^{\bf x}\{{\bf Y}_{t_{1}}\in F_{1},\cdots ,{\bf Y}_{t_{k}}\in F_{k}\}=\mathbb{P}^{0}\{{\bf Y}_{t_{1}}^{\bf x}\in F_{1},\cdots ,{\bf Y}_{t_{k}}^{\bf x}\in F_{k}\}\]
for $F_{i}$ Boresl sets. $\sharp$

This gives immediately the formula for the generator $A$ of an Ito diffusion.

\begin{equation}{\label{okst733}}\tag{7}\mbox{}\end{equation}

Theorem \ref{okst733}. (Oksendal \cite{oks}) Let ${\bf X}_{t}$ be the Ito diffusion $d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+ \boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}$. If $f\in C_{0}^{2}(\mathbb{R}^{n})$, i.e., $f\in C^{2}(\mathbb{R}^{n})$ and $f$ has compact support, then $f\in {\cal D}_{A}$ and
\begin{equation}{\label{okseq733}}
Af({\bf x})=\sum_{i}b_{i}({\bf x})\cdot\frac{\partial f}{\partial x_{i}}+\frac{1}{2}\sum_{i,j}(\boldsymbol{\sigma}\boldsymbol{\sigma}^{T})_{ij}({\bf x})\cdot
\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}. \sharp
\end{equation}

Example. The $n$-dimensional Brownian motion is of course the solution of the stochastic differential equation $d{\bf X}_{t}=d{\bf W}_{t}$, i.e., we have ${\bf b}={\bf 0}$ and $\boldsymbol{\sigma}=I_{n}$, the $n$-dimensional identity matrix, So the generator of ${\bf W}_{t}$ is
\[Af=\frac{1}{2}\sum\frac{\partial^{2}f}{\partial x_{i}}\]
for $f=f(x_{1},\cdots ,x_{n})\in C_{0}^{2}(\mathbb{R}^{n})$, i.e., $A=\frac{1}{2}\Delta$, where $\Delta$ is the Laplace operator. $\sharp$

Example. (The graph of Brownian motion). Let $W$ denote one-dimensional Brownian motion and let ${\bf X}=\left [\begin{array}{c}X^{1}\\ X^{2}\end{array}\right ]$ be the solution of the stochastic
differential equation
\[\left\{\begin{array}{ll}
dX^{1}=dt; & X^{1}(0)=t_{0}\\
dX^{2}=dW; & X^{2}(0)=x_{0}
\end{array}\right .\]
i.e.,
\[d{\bf X}={\bf b}dt+\boldsymbol{\sigma}d{\bf W};{\bf X}(0)=\left [\begin{array}{c}t_{0}\\ x_{0}\end{array}\right ],\]
with $latex {\bf b}=\left [\begin{array}{c}
1\\ 0\end{array}\right ]$ and $latex \boldsymbol{\sigma}=\left [
\begin{array}{c}
0\\1\end{array}\right ]$. In other words, ${\bf X}$ may be regarded as the graph of Brownian motion. The generator $A$ of ${\bf X}$ is given by
\[Af=\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}\mbox{ for }f=f(t,{\bf x})\in C_{0}^{2}(\mathbb{R}^{n}). \sharp\]

From now on we will, unless otherwise stated, let $A=A_{\bf X}$ denote the generator of the Ito diffusion ${\bf X}_{t}$. We let $L=L_{\bf X}$ denote the differential operator given by the right hand side of (\ref{okseq733}). From Theorem \ref{okst733} we know that $A_{\bf X}$ and $L_{\bf X}$ coincide on $C_{0}^{2}(\mathbb{R}^{n})$.

\begin{equation}{\label{okst741}}\tag{8}\mbox{}\end{equation}

Theorem \ref{okst741}. (Oksendal \cite{oks}(The Dynkin Formula) Let $f\in C_{0}^{2}(\mathbb{R}^{n})$. Suppose $T$ is a stopping time, $\mathbb{E}^{\bf x}[T]<\infty$. Then
\begin{equation}{\label{okseq741}}
\mathbb{E}^{\bf x}[f({\bf X}_{T})]=f({\bf x})+\mathbb{E}^{\bf x}\left [\int_{0}^{T}Af({\bf X}_{s})ds\right ]. \sharp
\end{equation}

Note that if $T$ is the first exit time of a bounded set, $E^{\bf x}[T]<\infty$, then (\ref{okseq741}) holds for any function $f\in C^{2}$.

The Characteristic Operator.

We now introduce an operator which is closely related to the generator $A$, but is more suitable in many situations.

Definition. Let $\{{\bf X}_{t}\}$ be an Ito diffusion. The characteristic operator ${\cal A}={\cal A}_{\bf X}$ of $\{{\bf X}_{t}\}$ is defined by
\begin{equation}{\label{okseq751}}
{\cal A}f({\bf x})=\lim_{U\downarrow {\bf x}}\frac{\mathbb{E}^{\bf x}[f({\bf X}_{T_{U}})]-f({\bf x})}{\mathbb{E}^{\bf x}[T_{U}]},
\end{equation}
where the $U$’s are open sets $U_{k}$ decreasing to the point ${\bf x}$, in the sense that $U_{k+1}\subseteq U_{k}$ and $\bigcap_{k}U_{k}=\{{\bf x}\}$, and $T_{U}=\inf\{t>0:{\bf X}_{t}\not\in U\}$ is the first exit time from $U$ for ${\bf X}_{t}$. The set of functions $f$ such that the limit (\ref{okseq751}) exists for all ${\bf x}\in \mathbb{R}^{n}$ and all $\{U_{k}\}$ is denoted by ${\cal D}_{\cal A}$. If $\mathbb{E}^{\bf x}[T_{U}]=\infty$ for all open sets $U$ containg ${\bf x}$, i.e., ${\bf x}\in U$, we define ${\cal A}f({\bf x})=0$. $\sharp$

It turns out that ${\cal D}_{A}\subseteq {\cal D}_{\cal A}$ and that $Af={\cal A}f$ for all $f\in {\cal D}_{A}$. We will only need that ${\cal A}_{\bf X}=L_{\bf X}$ coincide on $C^{2}$. To obtain this we first
clarify a property of exit times.

Definition. A point ${\bf x}\in \mathbb{R}^{n}$ is called a trap for $\{{\bf X}_{t}\}$ if
\[Q^{\bf x}(\{{\bf X}_{t}={\bf x}\mbox{ for all }t\})=1.\]
In other words, ${\bf x}$ is trap if and only if $T_{\{{\bf x}\}}=\infty$ a.s. $Q^{\bf x}$. For example, if ${\bf b}({\bf x}_{0})=\boldsymbol{\sigma}({\bf x}_{0})={\bf 0}$, then ${\bf x}_{0}$ is a trap for ${\bf X}_{t}$ (by strong uniqueness of ${\bf X}_{t}$). $\sharp$

Proposition. If ${\bf x}$ is not a trap for ${\bf X}_{t}$, then there exists an open set $U$ containing ${\bf x}$ such that $E^{\bf x}[T_{U}]<\infty$. $\sharp$

Theorem. (Oksendal \cite{oks}). Let $f\in C^{2}$. Then $f\in {\cal D}_{\cal A}$ and
\[{\cal A}f=\sum_{i}b_{i}\cdot\frac{\partial f}{\partial x_{i}}+\frac{1}{2}\sum_{i,j}(\boldsymbol{\sigma}\boldsymbol{\sigma}^{T})_{ij}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}.\]

Example. (Brownian motion on the unit circle). The characteristic operator of the process ${\bf Y}=\left [\begin{array}{c}Y^{1}\\ Y^{1}\end{array}\right ]$

satisfying the stochastic differential equations
\[\left\{\begin{array}{l}
dY^{1}=-\frac{1}{2}Y^{1}dt-Y^{2}dW\\
dY^{2}=-\frac{1}{2}Y^{2}dt+Y^{1}dW
\end{array}\right .\]
is
\[{\cal A}f(y_{1},y_{2})=\frac{1}{2}\left (y_{2}^{2}\cdot\frac{\partial^{2}f}{\partial y_{1}^{2}}-2y_{1}y_{2}\cdot\frac{\partial^{2}f}{\partial y_{1}\partial y_{2}}-y_{1}\cdot\frac{\partial f}{\partial y_{1}}
-y_{2}\cdot\frac{\partial f}{\partial y_{2}}\right ).\]
This is because $d{\bf Y}=-\frac{1}{2}{\bf Y}dt+{\bf KY}dW$, where
\[{\bf K}=\left [\begin{array}{cc}
0 & -1\\ 1 & 0\end{array}\right ]\]
so that $d{\bf Y}={\bf b}({\bf Y})dt+\boldsymbol{\sigma}({\bf Y})dW$ with
\[{\bf b}(y_{1},y_{2})=\left [\begin{array}{c}
-\frac{1}{2}y_{1}\\ -\frac{1}{2}y_{2}\end{array}\right ],
\boldsymbol{\sigma}(y_{1},y_{2})=\left [\begin{array}{c}
-y_{2}\\ y_{1}\end{array}\right ]\mbox{ and }
a=\frac{1}{2}\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}=\frac{1}{2}
\left [\begin{array}{cc}
y_{2}^{2} & -y_{1}y_{2}\\ -y_{1}y_{2} & y_{1}^{2}
\end{array}\right ].\]

Kolmogorov’s Backward equation. The Resolvent.

In the following we let ${\bf X}_{t}$ be an Ito diffusion in $\mathbb{R}^{n}$ with generator $A$. If we choose $f\in C_{0}^{2}(\mathbb{R}^{n})$ and $T=t$ in Dynkin’s formula (\ref{okseq741}) we see that
\[u(t,{\bf x})=\mathbb{E}^{\bf x}[f({\bf X}_{t})]\]
is differentiable with respect to $t$ and
\begin{equation}{\label{okseq811}}
\frac{\partial u}{\partial t}=\mathbb{E}^{\bf x}[Af({\bf X}_{t})].
\end{equation}
It turns out that the right hand side of (\ref{okseq811}) can be expressed in terms of $u$ also.

Theorem. (Oksendal \cite{oks}(Kolmogorov’s backward equation). Let $f\in C_{0}^{2}(\mathbb{R}^{n})$. Define
\begin{equation}{\label{okseq812}}
u(t,{\bf x})=E^{\bf x}[f({\bf X}_{t})].
\end{equation}
Then $u(t,\cdot )\in {\cal D}_{A}$ for each $t$ and
\begin{equation}{\label{okseq813}}
\frac{\partial u}{\partial t}=Au\mbox{ with }u(0,{\bf x})=f({\bf x})\mbox{ for }t>0, {\bf x}\in \mathbb{R}^{n},
\end{equation}
where the right hand side is to be interpreted as $A$ applied to the function ${\bf x}\mapsto u(t,{\bf x})$. Moreover if $w(t,{\bf x})\in C^{1,2}(\mathbb{R}\times \mathbb{R}^{2})$ is a bounded function
satisfying (\ref{okseq813}) then $w(t,{\bf x})=u(t,{\bf x})$, given by (\ref{okseq812}). $\sharp$

It is an important fact that the operator $A$ always has an inverse, at least if a positive multiple of the identity is substracted from $A$. This inverse can be expressed explicitly in terms of the diffusion ${\bf X}_{t}$.

Definition. For $\alpha >0$ and $g\in C_{b}(\mathbb{R}^{n})$ we define the resolvent operator $R_{\alpha}$ by
\[R_{\alpha}g({\bf x})=E^{\bf X}\left [\int_{0}^{\infty}e^{-\alpha t}\cdot g({\bf X}_{t})dt\right ].\]

Proposition. $R_{\alpha}g$ is a bounded continuous function. $\sharp$

Proposition. Let $g$ be a lower bounded, measurable function on $\mathbb{R}^{n}$ and define, for fixed $t\geq 0$, $u({\bf x})=\mathbb{E}^{\bf x}[g({\bf X}_{t})]$.

(i) If $g$ is lower semicontinuous, then $u$ is lower semicontinuous.

(ii) If $g$ is bounded and continuous, then $u$ is continuous. In other words, any Ito diffusion ${\bf X}_{t}$ is Feller-continuous. $\sharp$

Theorem. If $f\in C_{0}^{2}(\mathbb{R}^{n})$ then $R_{\alpha}(\alpha -A)f=f$ for all $\alpha >0$ and if $g\in C_{b}(\mathbb{R}^{n})$ then $R_{\alpha}g\in {\cal D}_{A}$ and $(\alpha -A)R_{\alpha}g=g$ for all $\alpha >0$. $\sharp$

With a little harder work we can obtain the following useful generalization of Kolmogorov’s backward equation.

Theorem. (Oksendal \cite{oks}(The Feynman-Kac Formula). Let $f\in C_{0}^{2}(\mathbb{R}^{n})$ and $q\in C(\mathbb{R}^{n})$. Assume that $q$ is lower bounded. Put
\begin{equation}{\label{okseq821}}
v(t,{\bf x})=E^{\bf x}\left [\exp\left (-\int_{0}^{t}q({\bf X}_{s})ds\right )\cdot f({\bf X}_{t})\right ].
\end{equation}
Then
\begin{equation}{\label{okseq822}}
\frac{\partial v}{\partial t}=Av-qv\mbox{ with }v(0,{\bf x})=f({\bf x})\mbox{ for }t>0, {\bf x}\in \mathbb{R}^{n}.
\end{equation}
Moreover, if $w(t,{\bf x})\in C^{1,2}(\mathbb{R}\times\mathbb{R}^{n})$ is bounded on $K\times \mathbb{R}^{n}$ for each compact $K\subset \mathbb{R}$ and $w$ solves $(\ref{okseq822})$, given by (\ref{okseq821}). $\sharp$

In Theorem \ref{okst733} we have seen that the generator of an Ito diffusion ${\bf X}_{t}$ given by
\[d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}\]
is a partial differential operator $L$ of the form
\begin{equation}{\label{okseq827}}
Lf=\sum a_{ij}\cdot\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}+\sum b_{i}\cdot\frac{\partial f}{\partial x_{i}}
\end{equation}
where $[a_{ij}]=\frac{1}{2}\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}$ and ${\bf b}=[b_{i}]$. It is natural to ask if one can also find processes whose generator has the form
\begin{equation}{\label{okseq828}}
Lf=\sum a_{ij}\cdot\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}+\sum b_{i}\cdot\frac{\partial f}{\partial x_{i}}-cf,
\end{equation}
where $c({\bf x})$ is a bounded and continuous function. If $c({\bf x})\geq 0$ the answer is yes and a process $\widetilde{{\bf X}}_{t}$ with generator (\ref{okseq828}) is obtained by {\bf killing} ${\bf X}_{t}$ at a certain (killing) time $\zeta$. By this we mean that there exists a random time $\zeta$ such that if we put $\widetilde{{\bf X}}_{t}={\bf X}_{t}$ if $t<\zeta$ and leave $\widetilde{{\bf X}}_{t}$ undefined if $t\geq\zeta$, then $\widetilde{{\bf X}}_{t}$ is also a strong Markov process and
\begin{equation}{\label{okseq8210}}
\mathbb{E}^{\bf x}[f(\widetilde{{\bf X}}_{t})]=\mathbb{E}^{\bf x}[f({\bf X}_{t}),t<\zeta ]=\mathbb{E}^{\bf x}\left [f({\bf X}_{t})\cdot\exp\left (-\int_{0}^{t}c({\bf X}_{s})ds\right )\right ]
\end{equation}
for all bounded continuous functions $f$ on $\mathbb{R}^{n}$. Let $v(t,{\bf x})$ denote the right hand side of (\ref{okseq8210}) with $f\in C_{0}^{2}(\mathbb{R}^{n})$. Then
\[\lim_{t\rightarrow 0}\frac{E^{\bf x}[f(\widetilde{{\bf X}}_{t})]-f({\bf x})}{t}=\left .\frac{\partial}{\partial t}v(t,{\bf x})\right |_{t=0}=\left . (Av-cv)\right |_{t=0}=Af({\bf x})-c({\bf x})f({\bf x})\]
by the Feynman-Kac formula. So the generator of $\widetilde{{\bf X}}_{t}$ is (\ref{okseq828}), as required. The function $c({\bf x})$ can be interpreted as the killing rate
\[c({\bf x})=\lim_{t\downarrow 0}\frac{1}{t}\cdot Q^{\bf x}(\mbox{${\bf X}_{0}$ is killed in the time interval $(0,t)$}).\]
Thus by applying such a killing procedure we can come from the special case $c=0$ in (\ref{okseq827}) to general case (\ref{okseq828}) with $c({\bf x})\geq 0$. Therefore, for many purposes it is enough to consider the equation (\ref{okseq827}).

The Martingale Problem.

If $d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma} ({\bf X}_{t})d{\bf W}_{t}$ is an Ito diffusion in $\mathbb{R}^{n}$ with generator $A$ and if $f\in C_{0}^{2}(\mathbb{R}^{n})$ then
\[f({\bf X}_{t})=f({\bf x})+\int_{0}^{t}Af({\bf X}_{s})ds+\int_{0}^{t}\nabla f^{T}({\bf X}_{s})\cdot\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}.\]
Define
\[M_{t}=f({\bf X}_{t})-\int_{0}^{t}Af({\bf X}_{s})ds\]
Then we see that
\[M_{t}=f({\bf x})+\int_{0}^{t}\nabla f^{T}({\bf X}_{s})\cdot\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}.\]
Since Ito integrals are martingales with respect to the $\sigma$-fields $\{{\cal F}_{t}^{(m)}\}$ we have for $s>t$
\[\mathbb{E}^{\bf x}[M_{s}|{\cal F}_{t}^{(m)}]=M_{t}.\]
It follows that
\[\mathbb{E}^{\bf x}[M_{s}|{\cal M}_{t}]=\mathbb{E}^{\bf x}[\mathbb{E}^{\bf x}[M_{s}|{\cal F}_{t}^{(m)}]|{\cal M}_{t}]=\mathbb{E}^{\bf x}[M_{t}|{\cal M}_{t}]=M_{t}\]
since $M_{t}$ is ${\cal M}_{t}$-measurable. We have proved the following result.

\begin{equation}{\label{okst831}}\tag{9}\mbox{}\end{equation}

Theorem \ref{okst831}. (Oksendal \cite{oks}) If ${\bf X}_{t}$ is an Ito diffusion in $\mathbb{R}^{n}$ with generator $A$, then for all $f\in C_{0}^{2}(\mathbb{R}^{n})$ the process
\[M_{t}=f({\bf X}_{t})-\int_{0}^{t}Af({\bf X}_{s})ds\]
is a martingale with respect to $\{{\cal M}_{t}\}$. $\sharp$

If we identify each $\omega\in\Omega$ with the function
\[\omega_{t}=\omega (t)={\bf X}_{t}^{\bf x}(\omega )\]
we see that the probability space $(\Omega ,{\cal M},Q^{\bf x})$ is identified with
\[((\mathbb{R}^{n})^{[0,\infty )},{\cal B},\widetilde{Q}^{\bf x})\]
where ${\cal B}$ is the Borel $\sigma$-filed on $(\mathbb{R}^{n})^{[0,\infty )}$. Thus, regarding the law of ${\bf X}_{t}^{\bf x}$ as a probability measure $\widetilde{Q}^{\bf x}$ on ${\cal B}$ we can formulate Theorem~\ref{okst831} as follows

Theorem. (Oksendal \cite{oks}) If $\widetilde{Q}^{\bf x}$ is the probability measure on ${\cal B}$ induced by the law $Q^{\bf x}$ of an Ito difussion ${\bf X}_{t}$, then for all $f\in C_{0}^{2}(\mathbb{R}^{n})$ the process
\[M_{t}=f({\bf X}_{t})-\int_{0}^{t}Af({\bf X}_{s})ds=f(\omega_{t})-\int_{0}^{t}Af(\omega_{s})ds\mbox{ for }\omega\in (\mathbb{R}^{n})^{[0,\infty )}\]

is a $\widetilde{Q}^{\bf x}$-martingale with respect to the Borel $\sigma$-fields ${\cal B}_{t}$ of $(\mathbb{R}^{n})^{[0,t]}$ for $t\geq 0$. In other words, the measure $\widetilde{Q}^{\bf x}$ solves the martingale problem for the differential operator $A$ in the following sense. $\sharp$

Definition. Let $L$ be a semi-elliptic differential operator of the form
\[L=\sum b_{i}\cdot\frac{\partial}{\partial x_{i}}+\sum_{i,j}a_{ij}\cdot\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\]

where the coefficients $b_{i}$ and $a_{ij}$ are locally bounded Borel measurable functions on $\mathbb{R}^{n}$. Then we say that a probability measure $\widetilde{P}^{\bf x}$ on $((\mathbb{R}^{n})^{[0,\infty )},{\cal B})$ solves the martingale problem for $L$ (starting at ${\bf x}$) if the process
\[M_{t}=f(\omega_{t})-\int_{0}^{t}Lf(\omega_{s})ds\mbox{ with }M_{0}=f({\bf x})\mbox{ a.s. }\widetilde{P}^{\bf x}\]

is a $\widetilde{P}^{\bf x}$-martingale with respect to ${\cal B}_{t}$ for all $f\in C_{0}^{2}(\mathbb{R}^{n})$. The martingale problem is called well-posed if there is a unique measure $\widetilde{P}^{\bf x}$ solving the martingale problem. $\sharp$

The argument of Theorem~\ref{okst831} actually proves that $\widetilde{Q}^{\bf x}$ solves the martingale problem or $A$ whenever ${\bf X}_{t}$ is a weak solution of the stochastic differential equation
\begin{equation}{\label{okseq834}}
d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}.
\end{equation}
Conversely, it can be proved that if $\widetilde{P}^{\bf x}$ solves the martingale problem for
\begin{equation}{\label{okseq835}}
L=\sum b_{i}\cdot\frac{\partial}{\partial x_{i}}+\frac{1}{2}\sum_{i,j}(\boldsymbol{\sigma}\boldsymbol{\sigma}^{T})_{ij}\cdot\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}
\end{equation}
starting at ${\bf x}$ for all ${\bf x}\in \mathbb{R}^{n}$, then there exists a weak solution ${\bf X}_{t}$ of the stochastic differential equation (\ref{okseq834}). Moreover, this weak solution ${\bf X}_{t}$ is a Markov process if and only if the martingale problem for $L$ is well-posed. Therefore, if the coefficients ${\bf b}$ and $\boldsymbol{\sigma}$ of (\ref{okseq834}) satisfy the conditions (\ref{okseq521}) and (\ref{okseq522}) of Theorem \ref{okst521}, we conclude that

Theorem {(Oksendal \cite{oks})} $\widetilde{Q}^{\bf x}$ is the unique solution of the martingale problem for the operator $L$ given by (\ref{okseq835}). $\sharp$

Lipschitz-continuity of the coefficients of $L$ is not necessary for the uniqueness of the martingale problem. For example, one of the particular results is that
\[L=\sum b_{i}\cdot\frac{\partial}{\partial x_{i}}+\sum_{i,j}a_{ij}\cdot\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\]
has a unique solution of the martingale problem if $[a_{ij}]$ is everywhere positive definite, $a_{ij}({\bf x})$ is continuous, ${\bf b}({\bf x})$ is measurable and there exists a constant $d$ such that $\parallel {\bf b}({\bf x})\parallel +\parallel {\bf a}({\bf x}\parallel^{1/2}\leq d\cdot (1+\parallel {\bf x}\parallel )$ for all ${\bf x}\in \mathbb{R}^{n}$.

When is an Ito Process a Diffusion.

The Ito formula gives that if we apply a $C^{2}$ finction $\boldsymbol{\phi} : U\subset \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ to an Ito process ${\bf X}_{t}$ the result $\boldsymbol{\phi} ({\bf X}_{t})$ is another Ito process. A natural qoestion is: If ${\bf X}_{t}$ is an Ito diffusion will $\boldsymbol{\phi}({\bf X}_{t})$ be an Ito diffusion too? The answer is no ingeneral, but is may be yes in some cases.

\begin{equation}{\label{{okst842}}\tag{10}\mbox{}\end{equation}

Proposition \ref{okst842}. { (Oksendal \cite{oks})}. An Ito process $d{\bf Y}_{t}={\bf v}d{\bf W}_{t}$ with ${\bf Y}_{0}={\bf 0}$ for ${\bf v}(t,\omega )\in {\cal V}_{{\cal H}}^{n\times m}$ coincides in law
with $n$-dimensional Brownian motion if and only if ${\bf v}{\bf v}^{T}(t,\omega )=I_{n}$ for a.a. $(t,\omega )$ with respect to $dt\times dP$, where $I_{n}$ is the $n$-dimensional identity matrix. $\sharp$

Proposition \ref{okst842} is a special case of the following result, which gives a necessary and sufficient condition for an Ito process to coincide in law with a given diffusion. We use the symbol “$\simeq$” for
“coincides in law with”.

\begin{equation}{\label{okst843}}\tag{11}\mbox{}\end{equation}

Theorem \ref{okst843}. Let ${\bf X}_{t}$ be an Ito diffusion given by
\[d{\bf X}_{t}={\bf b}({\bf X}_{t})dt+\boldsymbol{\sigma}({\bf X}_{t})d{\bf W}_{t}\mbox{ for }{\bf b}\in \mathbb{R}^{n},
\boldsymbol{\sigma}\in \mathbb{R}^{n\times m}, {\bf X}_{0}={\bf x},\]
and let ${\bf Y}_{t}$ be an Ito process given by
\[d{\bf Y}_{t}={\bf u}(t,\omega )dt+{\bf v}(t,\omega )d{\bf W}_{t}\mbox{ for }{\bf u}\in \mathbb{R}^{n},{\bf v}\in \mathbb{R}^{n\times m}, {\bf Y}_{0}={\bf x}.\]
Then ${\bf X}_{t}\simeq {\bf Y}_{t}$ if and only if $\mathbb{E}^{\bf x}[{\bf u}(t,\cdot )|{\cal N}_{t}]={\bf b}({\bf Y}_{t}^{\bf x})$ and $latex {\bf v}{\bf v}^{T}(t,\omega )=\boldsymbol{\sigma}
\boldsymbol{\sigma}^{T}({\bf Y}_{t}^{\bf x})$ for a.a. $(t,\omega )$ with respect to $dt\times dP$, where ${\cal N}_{t}$ is the $\sigma$-field generated by ${\bf Y}_{s}$ for $s\leq t$. $\sharp$

Corollary. Let $d{\bf Y}_{t}={\bf u}(t,\omega )dt+{\bf v}(t,\omega )d{\bf W}_{t}$ be an Ito process in $\mathbb{R}^{n}$. Then ${\bf Y}_{t}$ is a Brownina motion if and only if $\mathbb{E}^{\bf x}[{\bf u}(t,\cdot )|{\cal N}_{t}]={\bf 0}$ and ${\bf v}{\bf v}^{T}(t,\omega )=I_{n}$ for a.a. $(t,\omega )$. $\sharp$

Random Time Change.

Let $c(t,\omega )$ be an ${\cal F}_{t}$-adapted process. Define

\begin{equation}{\label{okseq851}}
\beta_{t}(\omega )=\beta (t,\omega )=\int_{0}^{t}c(s,\omega )ds.
\end{equation}
We will say that $\beta_{t}$ is a (random) {\bf time change} with time change rate $c(t,\omega )$. Note that $\beta_{t}(\omega )$ is also ${\cal F}_{t}$-adapted, and for each $\omega$ the map $\omega\mapsto\beta_{t}(\omega )$ is nondecreasing.Define $\alpha_{t}(\omega )=\alpha (t,\omega )$ by
\begin{equation}{\label{okseq852}}
\alpha_{t}=\inf\{s:\beta_{s}>t\}.
\end{equation}
Then $\alpha_{t}$ is a right-inverse of $\beta_{t}$, for each $\omega$,
\[\beta (\alpha (t,\omega ),\omega )=t\mbox{ for all }t\geq 0.\]
Moreover $t\mapsto\alpha_{t}(\omega )$ is right-continuous. If $c(t,\omega )>0$ for a.a. $(t,\omega )$ then $t\mapsto\beta_{t}(\omega )$ is strictly increasing, $t\mapsto\alpha_{t}(\omega )$ is continuous and $\alpha_{t}$ is also a left-inverse of $\beta_{t}$
\[\alpha (\beta (t,\omega ),\omega )=t\mbox{ for all }t\geq 0.\]
In general $\omega\mapsto\alpha_{t}(\omega )$ is an $\{{\cal F}_{s}\}$-stopping time for each $t$, since
\[\{\omega :\alpha_{t}(\omega )<s\}=\{\omega :t<\beta_{t}(\omega )\}\in {\cal F}_{s}.\]
We now ask the question: Suppose ${\bf X}_{t}$ is an Ito diffusion and ${\bf Y}_{t}$ an Ito process as in Theorem~\ref{okst843}. When does there exist a time change $\beta_{t}$ such that ${\bf Y}_{\alpha_{t}}\simeq {\bf X}_{t}$? (Note that $\alpha_{t}$ is only defined up to time $\beta_{\infty}$. If $\beta_{\infty}<\infty$ we interpret ${\bf Y}_{\alpha_{t}}\simeq {\bf X}_{t}$ to mean that ${\bf Y}_{\alpha_{t}}$ has the same law as ${\bf X}_{t}$ up to time $\beta_{\infty}$). Here is a partial answer.

Theorem. (Oksendal \cite{oks}). Let ${\bf X}_{t}$ and ${\bf Y}_{t}$ be as in Theorem \ref{okst843} and let $\beta_{t}$ be a time change with right inverse $\alpha_{t}$ as in (\ref{okseq851}) and (\ref{okseq852}) above, Assume that
\[{\bf u}(t,\omega )=c(t,\omega )\cdot {\bf b}({\bf Y}_{t})\mbox{ and }{\bf v}{\bf v}^{T}(t,\omega )=c(t,\omega )\cdot\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}({\bf Y}_{t})\]
for a.a. $(t,\omega )$. Then ${\bf Y}_{\alpha_{t}}\simeq {\bf X}_{t}$. $\sharp$

This result allows us to recognize time changes of Brownian motion

\begin{equation}{\label{okst852}}\tag{12}\mbox{}\end{equation}

Theorem \ref{okst852}. (Oksendal \cite{oks}) Let $d{\bf Y}_{t}={\bf v}(t,\omega )d{\bf W}_{t}$, ${\bf v}\in \mathbb{R}^{n\times m}$, ${\bf W}_{t}\in\mathbb{R}^{n}$, be an Ito integral in $\mathbb{R}^{n}$, ${\bf Y}_{0}={\bf 0}$ and assume that ${\bf v}{\bf v}^{T}(t,\omega)=c(t,\omega )I_{n}$ for some process $c(t,\omega )\geq 0$. Let $\alpha_{t}$ and $\beta_{t}$ be as in (\ref{okseq851}) and (\ref{okseq852}). Then ${\bf Y}_{\alpha_{t}}$ is an $n$-dimensional Brownian motion. $\sharp$

\begin{equation}{\label{oksc853}}\tag{13}\mbox{}\end{equation}

Corollary \ref{oksc853}. Let
\[dY_{t}=\sum_{i=1}^{n}v_{i}(t,\omega )dW^{i}_{t}(\omega )\mbox{ with }Y_{0}=0,\]
where ${\bf W}=(W^{1},\cdots ,W^{n})$ is a $n$-dimensional Brownian motion. Then $\widehat{W}_{t}\equiv Y_{\alpha_{t}}$ is a one-dimensional Brownian motion, where $\alpha_{t}$ is defined by $(\ref{okseq852})$ and
\[\beta_{t}=\int_{0}^{t}\left [\sum_{i=1}^{n}v_{i}^{2}(s,\omega )\right ]ds.\]

Corollary. Let $Y_{t}$ and $\beta_{t}$ be as in Corollary \ref{oksc853}. Assume that $\sum_{i=1}^{n}v_{i}^{2}(s,\omega )>0$ for a.a. $(s,\omega )$. Then there exists a Brownian motion $\widehat{W}_{t}$ such that $Y_{t}=\widehat{W}_{\beta_{t}}$. $\sharp$

Corollary. Let $c(t,\omega )\geq 0$ be given and define
\[d{\bf Y}_{t}=\int_{0}^{t}\sqrt{c(s,\omega )}d{\bf W}_{s},\]
where ${\bf W}_{s}$ is an $n$-dimensional Brownian motion, Then ${\bf Y}_{\alpha_{t}}$ is also an $n$-dimensional Brownain motion. $\sharp$

We now use this to prove that a time change of an Ito integral is again an Ito integral, but driven by a different Brownian motion $\widetilde{{\bf W}}_{t}$.

\begin{equation}{\label{okst857}}\tag{14}\mbox{}\end{equation}

Theorem \ref{okst857}. (Time change formula for Ito integrals). Suppose $c(s,\omega )$ and $\alpha (s,\omega )$ are $s$-continuous, $\alpha (0,\omega )=0$ for a.a. $\omega$ and that $E[\alpha_{t}]<\infty$. Let ${\bf W}_{s}$ be an $m$-dimensional Brownian motion and let ${\bf v}(s,\omega )\in {\cal V}_{\cal H}^{n\times m}$ be bounded and $s$-continuous. Define
\[\widetilde{{\bf W}}_{t}=\lim_{k\rightarrow\infty}\sum_{j}\sqrt{c(\alpha_{j},\omega )}\cdot\Delta {\bf W}_{\alpha_{j}}=\int_{0}^{\alpha_{j}}\sqrt{c(s,\omega )}d{\bf W}_{s}.\]
Then $\widetilde{{\bf W}}_{t}$ is an $m$-dimensional ${\cal F}_{\alpha_{t}}^{(m)}$-Briwnian motion (i.e. $\widetilde{{\bf W}}_{t}$ is a Brownian motion and $\widetilde{{bf W}}_{t}$ is a martingale with
respect to ${\cal F}_{\alpha_{t}}^{(m)}$) and
\[\int_{0}^{\alpha_{t}}{\bf v}(s,\omega )d{\bf W}_{s}=\int_{0}^{t}{\bf v}(\alpha_{s},\omega )\cdot\sqrt{\alpha^{\prime}_{s}(\omega )}d\widetilde{{\bf W}}_{s}\mbox{ a.s. }\mathbb{P},\]
where $\alpha^{\prime}_{s}(\omega )$ is the derivative of $\alpha (s,\omega )$ with respect to $s$, so that
\[\alpha^{\prime}_{s}(\omega )=\frac{1}{c(\alpha_{s},\omega )}\]
for a.a. $s\geq 0$ and a.a. $\omega\in\Omega$. $\sharp$

Example. (Brownian motion on the unit sphere in $\mathbb{R}^{n}$; $n\geq 3$). Let $S$ be the unit sphere of $\mathbb{R}^{n}$. Apply the function $\boldsymbol{\phi}:\mathbb{R}^{n}\setminus\{{\bf 0}\}\rightarrow S$ defined by
\[\boldsymbol{\phi}({\bf x})=\frac{{\bf x}}{\parallel {\bf x}\parallel}\mbox{ for }{\bf x}\in \mathbb{R}^{n}\setminus\{{\bf 0}\}\]
to $n$-dimensional Brownian motion ${\bf W}=(W^{1},\cdots ,W^{n})$. The result is a stochastic integral ${\bf Y}=(Y^{1},\cdots ,Y^{n})=\boldsymbol{\phi}({\bf W})$ which by Ito’s formula is given by
\[dY^{i}=\frac{\parallel {\bf W}\parallel^{2}-(W^{i})^{2}}{\parallel {\bf W}\parallel^{3}}dW^{i}-\sum_{j\neq i}\frac{W^{i}W^{j}}
{\parallel {\bf W}\parallel}dW^{j}-\frac{n-1}{2}\cdot\frac{W^{i}}{\parallel {\bf W}\parallel}dt\mbox{ for }i=1,\cdots ,n.\]
Hence
\[d{\bf Y}=\frac{1}{\parallel {\bf W}\parallel}\cdot\boldsymbol{\sigma}({\bf Y})d{\bf W}+\frac{1}{\parallel {\bf W}\parallel} \cdot {\bf b}({\bf Y})dt,\] where $\boldsymbol{\sigma}=[\sigma_{ij}]\in \mathbb{R}^{n\times n}$ with $\sigma_{ij}({\bf Y})=\delta_{ij}-Y^{i}Y^{j}$ for $1\leq i,j\leq n$ and ${\bf b}({\bf y})=-\frac{n-1}{2}\cdot\left [\begin{array}{c} y_{1}\\ \vdots\\ y_{n}\end{array}\right ]\in \mathbb{R}^{n}$, ($y_{1},\cdots y_{n}$ are the coordinates of ${\bf y}\in\mathbb{R}^{n}$). Now perform the following time change: Define
\[{\bf Z}_{t}(\omega )={\bf Y}_{\alpha_{t}(\omega )}(\omega )\]
where
\[\alpha_{t}=\frac{1}{\beta_{y}}\mbox{ and }\beta_{t}(\omega )=\int_{0}^{t}\frac{1}{\parallel {\bf W}\parallel^{2}}ds.\]
Then ${\bf Z}$ us again an Ito process and by Theorem~\ref{okst857}
\[d{\bf Z}=\boldsymbol{\sigma}({\bf Z})d{\bf W}+{\bf b}({\bf Z})dt.\]
Hence ${\bf Z}$ is a diffusion with characteristic operator
\[{\cal A}f({\bf y})=\frac{1}{2}\left (\Delta f({\bf y})-\sum_{i,j}y_{i}y_{j}\frac{\partial^{2}f}{\partial y_{i}\partial y_{j}}\right
)-\frac{n-1}{2}\cdot \sum_{i} y_{i}\cdot\frac{\partial f}{\partial y_{i}} \mbox{ with }\parallel {\bf y}\parallel =1.\]

Thus
$\boldsymbol{\phi}({\bf W})={\bf W}/\parallel {\bf W}\parallel$, after a suitable change of time scale, equal to a dissusion $Z$ living on the unit sphere $S$ of $\mathbb{R}^{n}$. Note that ${\bf Z}$ is invariant under orthogonal transformations in $\mathbb{R}^{n}$ (since ${\bf W}$ is). It is reasonable to call ${\bf Z}$ Brownian motion on the unit sphere $S$. }\end{Ex}

Example. (Harmonic and analytic functions). Let ${\bf W}=(W^{1},W^{2})$ be a two-dimensional Brownian motion. Let us investigate what happens if we apply a $C^{2}$ function $\boldsymbol{\phi}(x_{1},x_{2})=(u(x_{1},x_{2}),v(x_{1},x_{2}))$ to ${\bf W}$. Put ${\bf Y}=(Y^{1},Y^{2})=\boldsymbol{\phi}(W^{1},W^{2})$ and apply Ito’s formula.
\[dY^{1}=u’_{1}(W^{1},W^{2})dW^{1}+u’_{2}(W^{1},W^{2})dW^{2}+\frac{1}{2}(u”_{11}(W^{1},W^{2})+u”_{22}(W^{1},W^{2}))dt\]
and
\[dY^{2}=v’_{1}(W^{1},W^{2})dW^{1}+v’_{2}(W^{1},W^{2})dW^{2}+\frac{1}{2}(v”_{11}(W^{1},W^{2})+v”_{22}(W^{1},W^{2}))dt,\]
wheer $u’_{1}=\partial u/\partial x_{1}$ etc. So
\[d{\bf Y}={\bf b}(W^{1},W^{2})dt+\boldsymbol{\sigma}(W^{1},W^{2})d{\bf W},\]
where ${\bf b}=\frac{1}{2}\left [\begin{array}{c}\Delta u\\ \Delta v\end{array}\right ]$ and $latex \boldsymbol{\sigma}=\left [\begin{array}{cc}
u’_{1} & u’_{2}\\ v’_{1} & v’_{2}\end{array}\right ]=D_{\boldsymbol{\phi}}$ (the derivative of $\boldsymbol{\phi}$).
So ${\bf Y}=\boldsymbol{\phi}(W^{1},W^{2})$ is a martingale if (and, in fact, only if) $\boldsymbol{\phi}$ is harmonic, i.e., $\Delta\boldsymbol{\phi}={\bf 0}$. If $\boldsymbol{\phi}$ is harmonic, we get by Corollary \ref{oksc853} that
\[\boldsymbol{\phi}(W^{1},W^{2})=(\widetilde{W}_{\beta_{1}}^{1},\widetilde{W}_{\beta_{2}}^{2})\]
where $\widetilde{W}^{1}$ and $\widetilde{W}^{2}$ are two (not necessarily
independent) versions of one-dimensional Brownian motion, and
\[\beta_{1}(t,\omega )=\int_{0}^{t}\parallel\Delta u\parallel^{2}(W^{1},W^{2})ds\mbox{ and }\beta_{2}(t,\omega )=\int_{0}^{t}\parallel\Delta v\parallel^{2}(W^{1},W^{2})ds.\]
Since
\[\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}=\left [
\begin{array}{cc}
\parallel\Delta u\parallel^{2} & \Delta u\cdot\Delta v\\
\Delta u\cdot\Delta v & \parallel\Delta v\parallel^{2}\\
\end{array}\right ]\]
we see that if (in addition to $\Delta i=\Delta v=0$)
\begin{equation}{\label{okseq8519}}
\parallel\Delta u\parallel^{2}=\parallel\Delta v\parallel^{2}\mbox{ and }\Delta u\cdot\Delta v=0
\end{equation}
then
\[{\bf Y}_{t}={\bf Y}_{0}+\int_{0}^{t}\boldsymbol{\sigma}d{\bf W}\]
with $\boldsymbol{\sigma}\boldsymbol{\sigma}^{T}=\parallel\Delta u\parallel^{2}(W^{1},W^{2})\cdot I_{2}$ and ${\bf Y}_{0}=\boldsymbol{\phi}(W^{1}(0),W^{2}(0))$. Therefore, if we let
\[\beta_{t}(\omega )=\int_{0}^{t}\parallel\Delta u\parallel^{2}(W^{1},W^{2})ds\mbox{ and }\alpha_{t}=\frac{1}{\beta_{t}}\]
we obtain by Theorem~\ref{okst852} That ${\bf Y}_{\alpha_{t}}$ is a two-dimensional Brownian motion. Conditions (\ref{okseq8519}), in addition to $\Delta u=\Delta v=0$, are easily seen to be equivalent to requiring that the function $\boldsymbol{\phi}(x+iy)=\boldsymbol{\phi}(x,y)$ regarded as a complex unction is either analytic or conjugate analytic. Thus we have proved that $\boldsymbol{\phi}(W^{1},W^{2})$ is, after a change of time scale, again Brownian motion in the plabe if and only if $\boldsymbol{\phi}$ is either analytic or conjugate analytic. $\sharp$

The Girsanov Theorem.

Basically the Girsanov theorem says that if we change the drift coefficient of a given Ito process (with a nondegenerate diffusion coefficient), then the law of process will not change dramatically. In fact, the law of the new process will be absolutely continuous with respect to the law of original process and we can compute explicitly the Radon-Nikodym derivative.

\begin{equation}{\label{okst861}}\tag{15}\mbox{}\end{equation}

Theorem \ref{okst861}. (The L\'{e}vy characterization of Brownian motion).} Let ${\bf X}_{t}=(X^{(1)}_{t},\cdots ,X_{t}^{(n)})$ be a continuous stochastic process on a probability space $latex (\Omega ,{\cal
H},P)$ with values in $\mathbb{R}^{n}$. Then the following statements are equivalent

(i) ${\bf X}_{t}$ is an $n$-dimensional Brownian motion with respect to $\mathbb{P}$.

(ii) ${\bf X}_{t}$ is a martingale with respect to $P$ $($and with respect to its own filtration$)$ and $X^{(i)}_{t}\cdot X^{(j)}_{t}-\delta_{ij}\cdot t$ is a martingale with respect to $P$ $($and with respect to
its own filtration$)$ for all $i,j\in\{1,2,\cdots ,n\}$. $\sharp$

\begin{equation}{\label{okst863}}\tag{16}\mbox{}\end{equation}

Theorem \ref{okst863}. (The Girsanov’s Theorem I). Let $\{{\bf Y}_{s}\}_{0\leq s\leq t}$ be an $n$-dimensional Ito process of the form
\[d{\bf Y}_{s}=\boldsymbol{\mu}_{s}(\omega )ds+d{\bf W}_{s}\mbox{ with }{\bf Y}_{0}={\bf 0}\mbox{ for }s\leq t,\]
where $\{{\bf W}_{s}\}_{0\leq s\leq t}$ is $n$-dimensional Brownian motion. Put
\[\Gamma_{t}=\exp\left (-\int_{0}^{s}\boldsymbol{\mu}_{u}(\omega )d{\bf W}_{u}-\frac{1}{2}\int_{0}^{s}\parallel\boldsymbol{\mu}_{u}\parallel^{2}(\omega )du\right )\mbox{ for }s\leq t.\]
Assume that $\boldsymbol{\mu}_{u}(\omega )$ satisfies Novikov’s condition
\begin{equation}{\label{okseq867}}
\mathbb{E}_{P}\left [\exp\left (\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\mu}_{s}\parallel^{2}(\omega )ds\right )\right ]<\infty .
\end{equation}
Define the measure $\mathbb{Q}$ on $(\Omega ,{\cal F}_{t}^{(n)})$ by
\begin{equation}{\label{okseq868}}
dQ=\Gamma_{t}dP.
\end{equation}
Then $\{{\bf Y}_{s}\}_{0\leq s\leq t}$ is an $n$-dimensional Brownian motion with respect to the probability measure $Q$ for $s\leq t$. $\sharp$

The transform $\mathbb{P}\rightarrow\mathbb{Q}$ given by (\ref{okseq868}) is called the Girsanov transformation of measures. The Novikov condition (\ref{okseq867}) is sufficient to guarantee that $\{M_{s}\}_{0\leq s\leq t}$ is a martingale with respect to ${\cal F}_{s}^{(n)}$ and $P$. Actually, the result holds if we only assume that $\{M_{s}\}_{0\leq s\leq t}$ is a martingale. Note that since $\{M_{s}\}_{0\leq s\leq t}$ is a martingale we actually have that
\begin{equation}{\label{okseq869}}
M_{t}dP=M_{s}dP\mbox{ on }{\cal F}_{s}^{(n)}\mbox{ for }s\leq t.
\end{equation}
(The consistency condition) To see this, let $f$ be a bounded ${\cal F}_{s}^{(n)}$-measurable function. Then we have
\[\int_{\Omega}f\cdot M_{t}dP=\mathbb{E}[f\cdot M_{t}]=\mathbb{E}[\mathbb{E}[f\cdot M_{t}|{\cal F}_{s}]]=\mathbb{E}[f\cdot \mathbb{E}[M_{t}|{\cal F}_{s}]]=\mathbb{E}[f\cdot M_{s}]=\int_{\Omega}f\cdot M_{s}dP.\]

Theorem \ref{okst863} states that for all Borel sets $F_{1},\cdots ,F_{k} \subset\mathbb{R}^{n}$ and all $t_{1},t_{2},\cdots ,t_{k}\leq t$, $k=1,2,\cdots$ we have
\begin{equation}{\label{okseq8613}}
\mathbb{Q}\{{\bf Y}_{t_{1}}\in F_{1},\cdots ,{\bf Y}_{t_{k}}\in F_{k}\}=\mathbb{P}\{{\bf W}_{t_{1}}\in F_{1},\cdots ,{\bf W}_{t_{k}}\in F_{k}\}.
\end{equation}
An equivalent way is to say that $\mathbb{Q}\ll \mathbb{P}$ with Radon-Nikodym derivative
\[\frac{d\mathbb{Q}}{d\mathbb{P}}=\Gamma_{t}\mbox{ on }{\cal F}_{t}^{(n)}.\]
Note that $\Gamma_{t}>0$ a.s., so we also have that $\mathbb{P}\ll\mathbb{Q}$. Hence the two measures $\mathbb{P}$ and $\mathbb{Q}$ are equivalent. Therefore we get from (\ref{okseq8613})
\begin{align*}
\mathbb{P}\{{\bf Y}_{t_{1}}\in F_{1},\cdots ,{\bf Y}_{t_{k}}\in F_{k}\}>0 & \Leftrightarrow & Q\{{\bf Y}_{t_{1}}\in F_{1},\cdots ,{\bf Y}_{t_{k}}\in F_{k}\}>0\\
& \Leftrightarrow & \mathbb{P}\{{\bf W}_{t_{1}}\in F_{1},\cdots ,{\bf W}_{t_{k}}\in F_{k}\}>0
\end{align*}
for $t_{1},\cdots ,t_{k}\in [0,t]$.

\begin{equation}{\label{okst864}}\tag{17}\mbox{}\end{equation}

Theorem \ref{okst864}. )(The Girsanov’s Theorem II). Let $\{{\bf Y}_{s}\}_{0\leq s\leq t}$ be an $n$-dimensional Ito process of the form
\[d{\bf Y}_{s}=\boldsymbol{\mu}_{s}(\omega )ds+\boldsymbol{\sigma}_{s}(\omega )d{\bf W}_{s}\mbox{ for }s\leq t,\]

where $\{{\bf W}_{s}\}_{0\leq s\leq t}$ is an $m$-dimensional Brownian motion, $\boldsymbol{\mu}_{s}(\omega )\in\mathbb{R}^{n}$ and $latex \boldsymbol{\sigma}_{s}(\omega )\in
\mathbb{R}^{n\times m}$. Suppose there exist processes ${\bf u}_{s}(\omega )\in {\cal W}_{\cal H}^{m}$ and $\boldsymbol{alpha}_{s}\omega )\in {\cal W}_{\cal H}^{n}$ such that
\begin{equation}{\label{okseq8617}}
\boldsymbol{\sigma}_{s}(\omega )\cdot {\bf u}_{s}(\omega )=\boldsymbol{\mu}_{s}(\omega )-\boldsymbol{alpha}_{s}(\omega )
\end{equation}
and assuem that ${\bf u}_{s}(\omega )$ satisfies Novikov’s condition
\[\mathbb{E}_{P}\left [\exp\left (\frac{1}{2}\int_{0}^{t}\parallel {\bf u}_{s}\parallel^{2}(\omega )ds\right )\right ]<\infty\]
Put
\begin{equation}{\label{okseq8619}}
\Gamma_{s}=\exp\left (-\int_{0}^{s}{\bf u}_{u}(\omega )d{\bf W}_{u}-\frac{1}{2}\int_{0}^{s}\parallel {\bf u}_{u}\parallel^{2}(\omega )du\right )\mbox{ for }s\leq t.
\end{equation}
and
\begin{equation}{\label{okseq8620}}
d\mathbb{Q}=\Gamma_{t}(\omega )dP\mbox{ on }(\Omega ,{\cal F}_{t}^{(m)}).
\end{equation}
Then
\begin{equation}{\label{okseq8621}}
\widehat{{\bf W}}_{s}=\int_{0}^{s}{\bf u}_{u}(\omega )du+{\bf W}_{s}\mbox{ for }s\leq t
\end{equation}
is a Brownian motion with respect to $Q$ and in terms of $\widehat{{\bf W}}_{s}$ the process ${\bf Y}_{s}$ has the stochastic integral representation
\[d{\bf Y}_{s}=\boldsymbol{alpha}_{s}(\omega )ds+\boldsymbol{\sigma}_{s}\omega )d\widehat{{\bf W}}_{s}. \sharp\]

Note that if $n=m$ and $\boldsymbol{\sigma}\in\mathbb{R}^{n\times n}$ is invertible, then the process ${\bf u}_{s}(\omega )$ satisfying (\ref{okseq8617}) is given uniquely by
\[{\bf u}_{s}(\omega )=\boldsymbol{\sigma}^{-1}_{s}(\omega )(\boldsymbol{\mu}_{s}(\omega )-\boldsymbol{alpha}_{s}\omega )).\]
Finally we formulate a diffusion version

\begin{equation}{\label{okst865}}\tag{18}\mbox{}\end{equation}

Theorem \ref{okst865}. (The Girsanov’s Theorem III). Let ${\bf X}_{t}={\bf X}_{t}^{\bf x}$ and ${\bf Y}_{t}={\bf Y}_{t}^{\bf x}$ be an $n$-dimensional Ito diffussion and an $n$-dimensional
Ito process, respectively, of the form
\[d{\bf X}_{s}=\boldsymbol{\mu}({\bf X}_{s})ds+\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}\mbox{ with }{\bf X}_{0}={\bf 0}\mbox{ for }s\leq t\]
and
\[d{\bf Y}_{s}=(\boldsymbol{\gamma}_{s}(\omega )+\boldsymbol{\mu}({\bf Y}_{s}))ds+ \boldsymbol{\sigma}({\bf Y}_{s})d{\bf W}_{s}\mbox{ with } {\bf Y}_{0}={\bf
x}\mbox{ for }s\leq t,\]

where the functions $\boldsymbol{\mu}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ and $\boldsymbol{\sigma}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$ satisfy the conditions of Theorem \ref{okst521} and $\boldsymbol{\gamma}_{s}(\omega )\in {\cal W}_{\cal H}^{n}$, ${\bf x}\in \mathbb{R}^{n}$. Suppose there exist processes ${\bf u}_{s}(\omega ) \in {\cal W}_{\cal H}^{m}$ such that
\[\boldsymbol{\sigma}({\bf Y}_{s}){\bf u}_{s}(\omega )=\boldsymbol{\gamma}_{s}(\omega )\]
and assuem that ${\bf u}_{s}(\omega )$ satisfies Novikov’s condition
\[\mathbb{E}_{P}\left [\exp\left (\frac{1}{2}\int_{0}^{t}\parallel {\bf u}_{s}\parallel^{2}(\omega )ds\right )\right ]<\infty\]
Define $\Gamma_{s}$, $Q$ and $\widehat{{\bf W}}_{s}$ as in $(\ref{okseq8619})$, $(\ref{okseq8620})$ and $(\ref{okseq8621})$. Then
\[d{\bf Y}_{s}=\boldsymbol{\mu}({\bf Y}_{s})ds+\boldsymbol{\sigma}({\bf Y}_{s})d\widehat{{\bf W}}_{s}.\]
Therefore the $Q$-law of ${\bf Y}_{s}^{\bf x}$ is the same as the $P$-law of ${\bf X}_{s}^{\bf x}$ for $s\leq t$. $\sharp$

The Girsanov’s theorem III can be used to produce weak solutions of stochastic differential equations. To illustrate this, suppose ${\bf Y}_{s}$ is a known weak or strong solution to the equation
\[d{\bf Y}_{s}={\bf b}({\bf Y}_{s})ds+\boldsymbol{\sigma}({\bf Y}_{s})d{\bf W}_{s}\]

where ${\bf b}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $\boldsymbol{\sigma}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times m}$ and ${\bf W}_{s}$ is an $m$-dimensional Brownian motion. We wish to find a weak solution of a related equation
\begin{equation}{\label{okseq8630}}
d{\bf X}_{s}={\bf a}({\bf X}_{s})ds+\boldsymbol{\sigma}({\bf X}_{s})d{\bf W}_{s}
\end{equation}
where the drift function is changed to ${\bf a}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$. Suppose we can find a function ${\bf u}_{0}:\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ such that
\[\boldsymbol{\sigma}({\bf y})\cdot {\bf u}_{0}({\bf y})={\bf b}({\bf y})-{\bf a}({\bf y})\mbox{ for }{\bf y}\in \mathbb{R}^{n}.\]
(If $n=m$ and $\boldsymbol{\sigma}$ is invertiable we choose ${\bf u}_{0}=\boldsymbol{\sigma}^{-1}\cdot ({\bf b}-{\bf a})$.) Then if ${\bf u}_{s}(\omega )={\bf u}_{0}({\bf Y}_{s}(\omega ))$ satisfies Novikov’s conditions, we have, with $Q$ and $\widehat{{\bf W}}_{s}$ as in (\ref{okseq8620}) and (\ref{okseq8621}), that
\begin{equation}{\label{okseq8631}}
d{\bf Y}_{s}={\bf a}({\bf Y}_{s})ds+\boldsymbol{\sigma}({\bf Y}_{s})d\widehat{{\bf W}}_{s}.
\end{equation}
Thus we have found a Brownian motion $(\widehat{{\bf W}}_{s},Q)$ such that ${\bf Y}_{s}$ satisfies (\ref{okseq8631}). Therefore $({\bf Y}_{s},\widehat{{\bf W}}_{s})$ is a weak solution of (\ref{okseq8630}).

Example. Let ${\bf a}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be a bounded and measurable function. Then we can construct a weak solution ${\bf X}_{s}= {\bf X}_{s}^{\bf x}$ of the stochastic
differential equation
\begin{equation}{\label{okseq8632}}
d{\bf X}_{s}={\bf a}({\bf X}_{s})ds+d{\bf W}_{s}\mbox{ with }{\bf X}_{0}={\bf x}\in \mathbb{R}^{n}.
\end{equation}
We proceed according to the procedure above with $\boldsymbol{\sigma}=I$, ${\bf b}={\bf 0}$ and $d{\bf Y}_{s}=d{\bf W}_{s}$ with ${\bf Y}_{0}={\bf x}$. Choose ${\bf u}_{0}=\boldsymbol{\sigma}^{-1}\cdot ({\bf a}-{\bf a})=-{\bf a}$ and define
\[M_{s}=\exp\left (-\int_{0}^{s}{\bf u}_{0}({\bf Y}_{u})d{\bf W}_{u}-\frac{1}{2}\int_{0}^{s}\parallel {\bf u}_{0}({\bf Y}_{u})\parallel^{2}du\right )\]
i.e.
\[M_{s}=\exp\left (-\int_{0}^{s}{\bf a}({\bf W}_{u})d{\bf W}_{u}-\frac{1}{2}\int_{0}^{s}\parallel {\bf a}({\bf W}_{u})\parallel^{2}du\right ).\]
Fix $t<\infty$ and put $dQ=M_{t}dP$ on ${\cal F}_{t}^{(m)}$. Then
\[\widehat{{\bf W}}_{s}=-\int_{0}^{s}{\bf a}({\bf W}_{u})du+{\bf W}_{s}\]
is a Brownian motion with respect to $\mathbb{Q}$ for $s\leq t$ and
\[d{\bf W}_{s}=d{\bf Y}_{s}={\bf a}({\bf Y}_{s})ds+d\widehat{{\bf W}}_{s}.\]
Hence if we set ${\bf Y}_{0}={\bf x}$ the pair $({\bf Y}_{s},\widehat{{\bf W}}_{s})$ is a weak solution of (\ref{okseq8632}) for $s\leq t$. By weak uniqueness the $\mathbb{Q}$-law of ${\bf Y}_{s}={\bf W}_{s}$ coincides with the $\mathbb{P}$-law of ${\bf X}_{s}^{\bf x}$ so that
\[\mathbb{E}_{\mathbb{P}}[f_{1}({\bf X}_{t_{1}}^{\bf x})\cdots f_{k}({\bf X}_{t_{k}}^{\bf x})]=\mathbb{E}_{\mathbb{Q}}[f_{1}({\bf Y}_{t_{1}})\cdots f_{k}({\bf Y}_{t_{k}})]=
\mathbb{E}_{\mathbb{P}}[M_{T}\cdot f_{1}({\bf W}_{t_{1}})\cdots f_{k}({\bf W}_{t_{k}})]\] for all $f_{1},\cdots ,f_{k}\in C_{0}(\mathbb{R}^{n})$ and $t_{1},\cdots ,t_{k}\leq t$. } $\sharp$

Examples and Some Solution Methods.

Linear Stochastic Differential Equations.

Reducible Stochastic Differential Equations.

Some Explicitly Solvable SDEs.

Linear SDEs: Additive Noise.

Linear SDEs: Multiplicative Noise.

Reducible SDEs: Case I.

Reducible SDEs: Case 2.

Reducible SDEs: Case 3.

Reducible SDEs: Miscellaneous.

Linear SDEs with \(2\)-Dimensional Noise.

Multidimensional Stochastic Differential Equations.

The Existence and Uniqueness Results.

Diffusion Theory.

Diffusion Processes.

Strong Solutions as Diffusion Processes.

Diffusion Processes as Weak Solutions.

Vector Stochastic Differential Equations.

The Markov Property.

The Strong Markov Property.

The Generator of an Ito Diffusion.

The Characteristic Operator.

Kolmogorov’s Backward equation. The Resolvent.

The Martingale Problem.

When is an Ito Process a Diffusion.

Random Time Change.

The Girsanov Theorem.

Hsien-Chung Wu

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Examples and Some Solution Methods.

Linear Stochastic Differential Equations.

Reducible Stochastic Differential Equations.

Some Explicitly Solvable SDEs.

Linear SDEs: Additive Noise.

Linear SDEs: Multiplicative Noise.

Reducible SDEs: Case I.

Reducible SDEs: Case 2.

Reducible SDEs: Case 3.

Reducible SDEs: Miscellaneous.

Linear SDEs with \(2\)-Dimensional Noise.

Multidimensional Stochastic Differential Equations.

The Existence and Uniqueness Results.

Diffusion Theory.

Diffusion Processes.

Strong Solutions as Diffusion Processes.

Diffusion Processes as Weak Solutions.

Vector Stochastic Differential Equations.

The Markov Property.

The Strong Markov Property.

The Generator of an Ito Diffusion.

The Characteristic Operator.

Kolmogorov’s Backward equation. The Resolvent.

The Martingale Problem.

When is an Ito Process a Diffusion.

Random Time Change.

The Girsanov Theorem.

Hsien-Chung Wu

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