Stochastic Differential Equations 2

Friedrich Eduard Meyerheim (1808-1879) was a German painter.

 

The topics are as follows

• The Norms for Semimartingales
• Existence and Uniqueness of Solutions
• Stability of Stochastic Differential Equations
• The Markov Nature of Solutions
• Flows of Stochastic Differential Equations: Continuity and Differentiability
• Flows as Diffeomorphisms: The Continuous Case
• General Stochastic Exponentials and Linear Equations
• Flows as Diffeomorphisms: The General Case
• Miscellaneous.

A diffusion can be thought of as a strong Markov process (in $latex\mathbb{R}^{n}$) with continuous paths. Ito studied diffusions that could be represented as solutions of stochastic differential equations of the form
\[{\bf X}_{t}={\bf X}_{0}+\int_{0}^{t}\boldsymbol{\sigma}(X_{s})d{\bf W}_{s}+\int_{0}^{t}{\bf b}(X_{s})ds\]
where $latex{\bf W}$ is a $latexn$-dimensional Brownian motion and $latex\boldsymbol{\sigma}$ is an $latexn\times n$ matrix, $latex{\bf b}$ an $latexn$-vector, of appropriately smooth functions to ensure the existence
and uniqueness of solutions. This gives immediate intuitive meaning: If $latex\{{\cal F}_{t}\}_{t\geq 0}$ is the unerlying filtration for the Brownian motion $latex{\bf W}$, then for small $latex\epsilon >0$,
\begin{align*}
& \mathbb{E}\left [\left .X_{t+\epsilon}^{i}-X_{t}^{i}\right |{\cal F}_{t}\right ]=b^{i}(X_{t})\epsilon +o(\epsilon )\\
& \mathbb{E}\left [\left .(X_{t+\epsilon}^{i}-X_{t}^{i}-\epsilon b^{i}(X_{t}))\cdot (X_{t+\epsilon}^{j}-X_{t}^{j}-\epsilon b^{j}(X_{t}))\right |{\cal F}_{t}\right ]=(\boldsymbol{\sigma}\boldsymbol{\sigma}^{T})^{ij}(X{t})\cdot\epsilon _o(\epsilon ).
\end{align*}

Ito’s differential “d{\bf W}” was found to have other interpretations as well. In particular “d{\bf W}” can be thought of as “white noise” in statistical commuication theory, and thus if $latex\xi_{t}$ is white noise
at time $latext$, $latexW_{t}=\int_{0}^{t}\xi_{s}ds$, and equation which can be given a rigorous meaning using the theory of generalized functions. Finally it is now possible to consider semimartingale driving terms
(or “semimartingale noise”) and to study stochastic differential equations in full generality. Since “dW” and “dt” are semimartingale differentials, they are always included in the results as special cases.

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

The Norms for Semimartingales.

We have defined an $latex{\cal H}^{2}$ norm for semimartingale before as follows: if $latexX$ is a special semimartingale with $latexX_{0}=0$ and canonical decomposition $latexX=\bar{N}+\bar{A}$, then
\[\parallel X\parallel_{{\cal H}^{2}}=\parallel [\bar{N},\bar{N}]_{\infty}^{1/2}\parallel_{L^{2}}+\left |\!\left |\int_{0}^{\infty}|d\bar{A}_{a}|\right |\!\right |_{L^{2}}.\]
we now use an equivalent norm; to avoid confusion, we write $latex\underline{H}^{2}$ instead of $latex{\cal H}^{2}$; moreover we will define $latex\underline{H}^{p}$ for $latex1\leq p\leq\infty$.

We begin with a different norm on the space $latex{\bf D}$ (i.e. the space of adapted RCLL processes). For a process $latexH$ in $latex{\bf D}$ we define
\[H^{*}=\sup_{t}|H_{t}|\mbox{ and }\parallel H\parallel_{\underline{S}^{p}}=\parallel H^{*}\parallel_{L^{p}}.\]
Occasionally if $latexH$ is in $latex{\bf L}$ (adapted LCRL processes) we write $latex\parallel H\parallel_{\underline{S}^{p}}$ as well.

If $latexA$ is a semimartingale with paths of finite variation, a natural definition of a norm would be $latex\parallel A\parallel_{p}=\parallel\int_{0}^{\infty}|dA_{s}|\parallel_{L^{p}}$, where $latex|dA_{s}(\omega
)|$ denotes the total variation measure on $latex\mathbb{R}_{+}$ induced by $latexs\mapsto A_{s}(\omega )$. Since semimartingales do not in general have such nice paths, however, such a norm is not
appropriate. We will now assume that $latexZ$ is a semimartingale with $latexZ_{0}=0$ a.s. By the Bichteler-Dellacherire theorem in Theorem \ref{prot322}, we know there exists at least one decomposition $latexZ=N+A$ with $latexN$ a local martingale and $latexA$ an adapted, RCLL process with paths of finite variation (also $latexN_{0}=A_{0}=0$ a.s.). For $latex1\leq p\leq\infty$, we set
\[j_{p}(N,A)=\left |\!\left |[N,N]_{\infty}^{1/2}+\int_{0}^{\infty}|dA_{s}|\right |\!\right |_{L^{p}}.\]

Definition. Let $latexZ$ be a semimartingale. For $latex1\leq p\leq\infty$, we define
\[\parallel Z\parallel_{\underline{H}^{p}}=\inf_{Z+N+A}j_{p}(N,A)\]
where infimum is taken over all possible decompositions $latexZ=N+A$ where $latexN$ is a local martingale, $latexA\in {\bf D}$ with paths of finite variation on compacts, and $latexA_{0}=N_{0}=0$. $latex\sharp$

The Corollary \ref{proc51} below shows that this norm generalizes the $latex\underline{H}^{p}$ norm for local martingales.

\begin{equation}{\label{prot51}}\tag{1}\mbox{}\end{equation}

Proposition \ref{prot51}. (Protter \cite{pro}) Let $latexZ$ be a semimartingale with $latexZ_{0}=0$. Then
\[\parallel [Z,Z]_{\infty}^{1/2}\parallel_{L^{p}}\leq\parallel Z\parallel_{\underline{H}^{p}}\]
for $latex1\leq p\leq\infty$.

Proof. Let $latexZ=M+A$ with $latexM_{0}=A_{ 0}=0$ be a decomposition of $latexZ$. Then

\begin{align*}
[Z,Z]_{\infty}^{1/2} & \leq [M,M]_{\infty}^{1/2}+[A,A]_{\infty}^{1/2}\\
& =[M,M]_{\infty}^{1/2}+\left (\sum_{s}(\Delta A_{s})^{2}\right )^{1/2}\\
& \leq [M,M]_{\infty}^{1/2}+\sum_{s}|\Delta A_{s}|\\
& \leq [M,M]_{\infty}^{1/2}+\int_{0}^{\infty}|dA_{s}|,
\end{align*}
where the equality above holds because $latexA$ is a quadratic pure jump semimartingale. Taking $latexL^{p}$ norms yields $latex\parallel [Z,Z]_{\infty}^{1/2}\parallel_{L^{p}}\leq j_{p}(M,A)$ and the result follows. $latex\blacksquare$

\begin{equation}{\label{proc51}}\tag{2}\mbox{}\end{equation}

Corollary \ref{proc51}. If $latexZ$ is a local martingale with $latexZ_{0}=0$, then $latex\parallel Z\parallel_{\underline{H}^{p}}=\parallel [Z,Z]_{\infty}^{1/2}\parallel_{L^{p}}$.

Proof. Since $latexZ$ is a local martingale, we have that $latexZ=Z+0$ is a decomposition of $latexZ$. Therefore
\[\parallel Z\parallel_{\underline{H}^{p}}\leq j_{p}(Z,0)=\parallel [Z,Z]_{\infty}^{1/2}\parallel_{L^{p}}.\]
By Proposition \ref{prot51} we have $latex\parallel [Z,Z]_{\infty}^{1/2}\parallel_{L^{p}}\leq\parallel Z\parallel_{\underline{H}^{p}}$, hence we have equality. $latex\blacksquare$

\begin{equation}{\label{prot52}}\tag{3}\mbox{}\end{equation}

Proposition \ref{prot52}. (Protter \cite{pro}). For $latex1\leq p<\infty$ there exists a constant $latexc_{p}$ such that for any semimartingale $latexZ$ with $latexZ_{0}=0$, $latex\parallel Z\parallel_{\underline{S}^{p}}\leq c_{p}\cdot\parallel Z\parallel_{\underline{H}^{p}}$. $latex\sharp$

Corollary. On the space of semimartingales, the $latex\underline{H}^{p}$ norm is stronger than the $latex\underline{S}^{p}$ norm for $latex1\leq p<\infty$. $latex\sharp$

\begin{equation}{\label{prot53}}\tag{4}\mbox{}\end{equation}

Proposition \ref{prot53}. (Protter \cite{pro})(Emery’s Inequality). Let $latexZ$ be a semimartingale, $latexH\in {\bf L}$, and $latex1/p+1/q=1/r$ for $latex1\leq p\leq\infty$ and $latex1\leq q\leq\infty$. Then
\[\left |\!\left |\int_{0}^{\infty}H_{s}dZ_{s}\right |\!\right |_{\underline{H}^{r}}\leq\parallel H\parallel_{\underline{S}^{p}}\cdot\parallel Z\parallel_{\underline{H}^{q}}. \sharp\]

For a process $latexX\in {\bf D}$ an a stopping time $latexT$, recall that
\[X^{T}=X_{t}\cdot I_{[0,T)}+X_{T}\cdot I_{[T,\infty )}\mbox{ and }X^{T-}=X_{t}\cdot I_{[0,T)}+X_{T-}\cdot I_{[T,\infty )}.\]
Recall that a property $latex\pi$ is said to hold locally for a process $latexX$ if $latexX^{T_{n}}\cdot I_{\{T^{n}>0\}}$ has property $latex\pi$ for each $latexn$, where $latexT^{n}$ is a sequence of stopping times tending to $latex\infty$ a.s. If the process $latexX$ is zero at zero, i.e., $latexX_{0}=0$ a.s., then the property $latex\pi$ is said to hold pre-locally if $latexZ^{T^{n}-}$ has property for each $latexn$.

Definition. A process $latexX$ is locally in $latex\underline{S}^{p}$ (resp. $latex\underline{H}^{p}$ ) if there exist stopping times $latex\{T^{n}\}_{n\geq 1}$ increasing to $latex\infty$ a.s. such that $latexX^{T^{n}}\cdot I_{\{T^{n}>0\}}$ is in $latex\underline{S}^{p}$ $latex($resp. $latex\underline{H}^{p})$ for each $latexn$ and $latex1\leq p\leq\infty$. If $latexX_{0}=0$ then $latexX$ is said to be prelocally in $latex\underline{S}^{p}$ (resp. $latex\underline{H}^{p}$) if $latexX^{T^{n}-}$ is in $latex\underline{S}^{p}$ (resp. $latex\underline{H}^{p}$) for each $latexn$. $latex\sharp$

Proposition.  (Protter \cite{pro}). Let $latexZ$ be a semimartingale with $latexZ_{0}=0$. Then $latexZ$ is prelocally in $latex\underline{H}^{p}$ for $latex1\leq p\leq\infty$. $latex\sharp$

Definition. Let $latexZ$ be a semimartingale in $latex\underline{H}^{\infty}$ and let $latex\alpha >0$. A finite sequence of stopping times $latex0=T_{0}\leq T_{1}\leq\cdots\leq T_{k}$ is said to $latex\alpha$-slice $latexZ$ if
\[Z=Z^{T_{k}-}\mbox{ and }\parallel (Z-Z^{T_{i}})^{T_{i+1}-}\parallel_{\underline{H}^{\infty}}\leq\alpha\]
for $latex0\leq i\leq k-1$. If such a sequence of stopping times exists, we say $latexZ$ is $latex\alpha$-sliceable, and we write $latexZ\in {\cal S}(\alpha )$. $latex\sharp$

\begin{equation}{\label{prot55}}\tag{5}\mbox{}\end{equation}

Proposition \ref{prot55} (Protter \cite{pro}).  Let $latexZ$ be a semimartingale with $latexZ_{0}=0$ a.s. We have the following properties.

(i) For $latex\alpha >0$, if $latexZ\in {\cal S}(\alpha )$ then for every stopping time $latexT$, $latexZ^{T}\in {\cal S}(\alpha )$ and $latexZ^{T-}\in {\cal S}(2\alpha )$.

(ii) For every $latex\alpha >0$, there exists an arbitrarily large stopping time $latexT$ such that $latexZ^{T-}\in {\cal S}(\alpha )$. $latex\sharp$

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

Existence and Uniqueness of Solutions.

Recall that a process $latexH$ is in $latex{\bf L}$ if it has LCRL paths and is adapted.

\begin{equation}{\label{prot56}}\tag{6}\mbox{}\end{equation}

Proposition \ref{prot56}. (Protter \cite{pro}). Let $latexZ$ be a semimartingale with $latexZ_{0}=0$ and let $latexf:\mathbb{R}_{+}\times \Omega\times \mathbb{R}\rightarrow \mathbb{R}$ be such that

• for fixed $latexx$, $latex(t,\omega )\mapsto f(t,\omega ,x)$ is in $latex{\bf L}$;
• for each $latex(t,\omega )$, $latex|f(t,\omega ,x)-f(t,\omega ,y)|\leq K(\omega )\cdot |x-y|$ for some finite random variable $latexK$.

Let $latexX_{0}$ be finite and $latex{\cal F}_{0}$-measurable. Then the equation
\[X_{t}=X_{0}+\int_{0}^{t}f(s,\cdot ,X_{s-})dZ_{s}\]
admits a solution. The solution is unique and it is a semimartingale. $latex\sharp$

Definition. A function $latexf:\mathbb{R}_{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is {\bf Lipschitz} if there exists a constant $latexk$ such that

• \(|f(t,{\bf x})-f(t,{\bf y})|\leq k\cdot |{\bf x}-{\bf y}|\) for each $latext\in \mathbb{R}_{+}$;
• \(t\mapsto f(t,{\bf x})\) is right-continuous with left limits for each $latex{\bf x}\in \mathbb{R}^{n}$.

$f$ is said to be autonomous if $latexf(t,{\bf x})=f({\bf x})$ for all $latext\geq 0$. $latex\sharp$

Definition. A function $latexf:\mathbb{R}_{+}\times\Omega\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ is random Lipschitz if $latexf$ satisfies conditions $latex(i)$ and $latex(ii)$ of Proposition \ref{prot56}. $latex\sharp$

Let $latex{\bf D}^{n}$ denote the space of processes $latex{\bf X}=(X^{1},\cdots ,X^{n})$ where each $latexX^{i}\in {\bf D}$ for $latex1\leq i\leq n$.

Definition. An operator $latexF$ from $latex{\bf D}^{n}$ into $latex{\bf D}$ is said to be process Lipschitz if for any $latex{\bf X},{\bf Y}\in {\bf D}^{n}$, the following two conditions are satisfied

• for any stopping time $latexT$, $latex{\bf X}^{T-}={\bf Y}^{T-}$ implies $latexF({\bf X})^{T-}=F({\bf Y})^{T-}$;
• there exists an adapted process $latexK\in {\bf L}$ such that $latex\parallel F({\bf X})_{t}-F({\bf Y})_{t}\parallel\leq K_{t}\cdot\parallel {\bf X}_{t}-{\bf Y}_{t}\parallel$. $latex\sharp$

Definition. An operator $latexF$ mapping $latex{\bf D}^{n}$ to $latex{\bf D}$ is functional Lipschitz if for any $latex{\bf X},{\bf Y}\in {\bf D}^{n}$ the following two conditions are satisfied

• for any stopping time $latexT$, $latex{\bf X}^{T-}={\bf Y}^{T-}$ implies $latexF({\bf X})^{T-}=F({\bf Y})^{T-}$;
• there exists an increasing process $latexK=\{K_{t}\}_{t\geq 0}$ such that $latex\parallel F({\bf X})_{t}-F({\bf Y})_{t}\parallel\leq K_{t}\cdot\parallel {\bf X}-{\bf Y}\parallel_{t}^{*}$ a.s. for each $latext\geq 0$. $latex\sharp$

Note that if $latexg(t,{\bf x})$ is a Lipschitz function, then $latexf(t,{\bf x})=g(t-,{\bf x})$ is random Lipschitz. A Lipschitz, or a random Lipschitz function induces a process Lipschitz operator, and if an operator is
process Lipschitz, then it is also functional Lipschitz. An autonomous function with a bounded derivative is Lipschitz by the Mean Value Theorem. If a function $latexf$ has a continuous but not bounded derivative,
$f$ will be locally Lipschitz.

Let $latexA=\{A_{t}\}_{t\geq 0}$ be continuous and adapted. Then a linear coefficient such as $latexf(t,\omega ,{\bf x})=A_{t}(\omega ){\bf x}$ is an example of a process Lipschitz coefficient. A functional Lipschitz operator $latexF$ will typically be of the form $latexF(X)=f(t,\omega ;X_{s},s\leq t)$, where $latexf$ is defined on $latex[0,t]\times\Omega\times D[0,t]$ for each $latext\geq 0$; here $latexD[0,t]$ denotes the space of RCLL functions defined on $latex[0,t]$. Another example is a generalization of the coefficients
\[F(X)_{t}=\int_{0}^{t}g(u,\omega ,X_{u})\mu (\omega ,du)\]
for a random signed measure $latex\mu$ and a bounded Lipschitz function $latexg$ with constant $latexC(\omega )$. In this case, the Lipschitz process $latexF$ is given by $latexK_{t}(\omega )=C(\omega )\cdot\parallel\mu (\omega )_{t}\parallel$, where $latex\parallel\mu (\omega )_{t}\parallel$ denotes the total mass of the measure $latex\mu (\omega ,du)$ on $latex[0,t]$.

Proposition. (Protter \cite{pro}). Let $latex1\leq p<\infty$, let $latexJ\in\underline{S}^{p}$, let $latexF$ be functional Lipschitz and suppose $latexF(0)=0$, and that $latex\sup_{t}|K_{t}(\omega )|\leq k$ a.s. Let $latexZ$ be a semimartingale in $latex\underline{H}^{\infty}$ such that $latex\parallel Z\parallel_{\underline{H}^{\infty}}\leq 1/2\cdot c_{p}\cdot k$. Then the equation
\[X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s}\]
has a solution in $latex\underline{S}^{p}$, it is unique, and moreover
\[\parallel X\parallel_{\underline{S}^{p}}\leq 2\cdot\parallel J\parallel_{\underline{S}^{p}}.\]

Proof. Define $latex\Lambda :\underline{S}^{p}\rightarrow\underline{S}^{p}$ by
\[\Lambda (X)_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s}.\]
Then by Propositions~\ref{prot52} and ~\ref{prot53}, the operator is $latex\frac{1}{2}$-Lipschitz, and the fixed point theorem gives existence and uniqueness. Indeed
\begin{align*}
\parallel X\parallel_{\underline{S}^{p}} & \leq\parallel J\parallel_{\underline{S}^{p}}+\left |\!\left |\int F(X)_{s-}dZ_{s}\right |\!\right |_{\underline{S}^{p}}\\
& \leq\parallel J\parallel_{\underline{S}^{p}}+c_{p}\cdot\parallel F(X)\parallel_{\underline{S}^{p}}\cdot\parallel Z \parallel_{\underline{H}^{\infty}}\\
& \leq\parallel J\parallel_{\underline{S}^{p}}+\frac{1}{2k}\cdot\parallel F(X)\parallel_{\underline{S}^{p}}.
\end{align*}
Since $latex\parallel F(X)\parallel_{\underline{S}^{p}}=\parallel F(X)-F(0)\parallel_{\underline{S}^{p}}$, we have $latex\parallel X\parallel_{\underline{S}^{p}}\leq\parallel J\parallel_{\underline{S}^{p}}+
\frac{1}{2}\cdot\parallel X\parallel_{\underline{S}^{p}}$, which yields the estimate. $latex\blacksquare$

Proposition. (Protter \cite{pro}). Let $latexq\leq p<\infty$, let $latexJ\in\underline{S}^{p}$, let $latexF$ be functional Lipschitz with $latexF(0)=0$ and $latex\sup_{t}|K_{t}(\omega )|\leq k<\infty$ a.s. Let $latexZ$ be a semimartingale such that $latexZ\in {\cal S}(1/2\cdot c_{p}\cdot k)$. Then the equation
\[X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s}\]
has a solution in $latex\underline{S}^{p}$, it is unique, and moreover $latex\parallel X\parallel_{\underline{S}^{p}}\leq C(k,Z)\cdot\parallel Z\parallel_{\underline{S}^{p}}$, where $latexC(k,Z)$ is a constant depending only on $latexk$ and $latexZ$. $latex\sharp$

\begin{equation}{\label{prot57}}\tag{7}\mbox{}\end{equation}

Proposition \ref{prot57} (Protter \cite{pro}). Given a vector of semimartingales $latex{\bf Z}=(Z^{1},\cdots ,Z^{d})$ with $latex{\bf Z}_{0}={\bf 0}$, processes $latexJ^{i}\in {\bf D}$ for $latex1\leq i\leq n$, and operators $latexF_{j}^{i}$ which are functional Lipschitz for $latex1\leq i\leq n$ and $latex1\leq j\leq d$, then the system of equations
\[X_{t}^{i}=J_{t}^{i}+\sum_{j=1}^{d}\int_{0}^{t}F_{j}^{i}({\bf X})_{s-}dZ_{s}^{j}\]
for $latex1\leq i\leq n$ has a solution in $latex{\bf D}^{n}$, and it is unique. Moreover if $latex\{J^{i}\}_{i\leq n}$ is a vector of semimartingales, then so is $latex\{X^{i}\}_{i\leq n}$. $latex\sharp$

We have already met the stochastic exponential equation ($Z_{0}=0$)
\[X_{t}=X_{0}+\int_{0}^{t}X_{s-}dZ_{s},\]
where we obtained a formula for its solution
\[X_{t}=X_{0}\cdot\exp\left (Z_{t}-\frac{1}{2}[Z,Z]_{t}\right )\cdot\prod_{0<s\leq t}(1+\Delta Z_{s})\cdot\exp\left (-\Delta Z_{s}+\frac{1}{2}(\Delta Z_{s})^{2}\right ).\]
The uniqueness of this solution is a consequence of Proposition \ref{prot57}, or of Proposition \ref{prot56}.

A traditional way to show the existence and uniqueness of solutions of ordinary differential equation is the Picard iteration method; one might well wonder if Picard-type iterations converge in the case of stochastic differential equations. As it turns out, the following result is quite usefu

Proposition. (Protter \cite{pro}). Let the hypotheses of Proposition~\ref{prot57} be satisfied, and in addition let $latex(X^{0})^{i}=H^{i}$ be processes in $latex{\bf D}$ for $latex1\leq i\leq n$, and define inductively
\[(X_{t}^{m+1})^{i}=J_{t}^{i}+\sum_{i=1}^{d}\int_{0}^{t}F_{j}^{i}({\bf X}^{m})_{s-}dZ_{s}^{j}\]
and let $latex{\bf X}$ be the solution of
\[X_{t}^{i}=J_{t}^{i}+\sum_{j=1}^{d}\int_{0}^{t}F_{j}^{i}({\bf X})_{s-}dZ_{s}^{j}\mbox{ for }1\leq i\leq n.\]

Then $latex{\bf X}^{m}$ converges to $latex{\bf X}$ in ucp. $latex\sharp$

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

Stability of Stochastic Differential Equations.

Since one is never exactly sure of the accuracy of a proposed model, it is important to know how robust the model is; that is, if one perturbs the model a bit, how large are the resulting changes? Stochastic
differential equations are stable with respect to perturbations of the coefficients, or of the initial conditions. Perturbations of the differentials, however, is a more delicate matter: One must perturb the
differentials in the right way to have stability. Not surprisingly, an $latex\underline{H}^{p}$ perturbation is the right kind of perturbation. Here we will be concerned with equations of the form
\begin{equation}{\label {proeq5*n}}
X_{t}^{n}=J_{t}^{n}+\int_{0}^{t}F^{n}(X^{n})_{s-}dZ_{s}^{n}
\end{equation}
and
\begin{equation}{\label {proeq5*}}
X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s},
\end{equation}
where $latexJ^{n},J$ are in $latex{\bf D}$, $latexZ^{n},Z$ are semimartingales, and $latexF^{n},F$ are functional Lipschitz with Lipschitz processes $latexK_{n},K$ resppectively. We will assume that the Lipschitz processes $latexK_{n},K$ are each uniformly bounded by the same constant, and that the semimartingale differentials $latexZ^{n},Z$ are always zero at $latex0$; that is, $latexZ_{0}^{n}=0$ a.s. for $latexn\geq 1$ and $latexZ_{0}=0$ a.s. We say that a functional Lipschitz operator $latexF$ is {\bf bounded} if for all $latexH\in {\bf D}$, there exixts a nonrandom constant $latexc<\infty$ such that $latexF(H)^{*}<c$.

\begin{equation}{\label{prot59}}\mbox{}\end{equation}

Proposition \ref{prot59}. (Protter \cite{pro}). Let $latexJ,J^{n}\in {\bf D}$, $latexZ,Z^{n}$ be semimartingales and $latexF,F^{n}$ be functional Lipschitz with constants $latexK,K_{n}$ respectively. Assume

• \(J,J^{n}\) are in $latex\underline{S}^{2}$ (resp. $latex\underline{H}^{2}$) and $latex\lim_{n\rightarrow\infty}J^{n}=J$ in $latex\underline{S}^{2}$ (resp. $latex\underline{H}^{2}$);
• \(F^{n}\) are all bounded by the same constant $latexc$, and $latex\lim_{n\rightarrow\infty}F^{n}(X)=F(X)$ in $latex\underline{S}^{2}$, where $latexX$ is the solution of (\ref{proeq5*});
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s. $latex(a$ is not random$)$; $latexZ\in {\cal S}(1/(2\cdot\sqrt{8}\cdot a))$; $latex\{Z^{n}\}_{n\geq 1}$ are in $latex\underline{H}^{2}$, and $latex\lim_{n\rightarrow\infty}Z^{n}=Z$ in $latex\underline{H}^{2}$.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ in $latex\underline{S}^{2}$ (resp. in $latex\underline{H}^{2})$, where $latexX^{n}$ is the solution of $latex(\ref{proeq5*n}$) and $latexX$ is the solution of $latex(\ref{proeq5*})$. $latex\sharp$

\begin{equation}{\label{prot510}}\tag{8}\mbox{}\end{equation}

Proposition \ref{prot510}. Let $latexJ,J^{n}\in {\bf D}$; $latexZ$ be a semimartingale, $latexF,F^{n}$ be functional Lipschitz with constants $latexK,K_{n}$ respectively, and let $latexX^{n},X$ be the unique solutions of equations (\ref{proeq5*n}) and (\ref{proeq5*}). Assume

• \(J,J^{n}\) are in $latex\underline{S}^{2}$ and $latex\lim_{n\rightarrow\infty}J^{n}=J$ in $latex\underline{S}^{2}$;
• \(\lim_{n\rightarrow\infty}F^{n}(X)=F(X)\) in $latex\underline{S}^{2}$, where $latexX$ is the solution of $latex(\ref{proeq5*})$;
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s. for a non-random constant $latexa$, and $latexZ\in {\cal S}(1/(2\cdot\sqrt{8}\cdot a))$.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ in $latex\underline{S}^{2}$ where $latexX^{n}$ is the solution of (\ref{proeq5*n}) and $latexX$ is the solution of $latex(\ref{proeq5*})$. $latex\sharp$

We now wish to localize the results of Propositions \ref{prot59} and \ref{prot510} so that they hold for general semimartingales and exogeneous processes $latexJ^{n},J$.

Definition. Processes $latexM^{n}$ are said to converge locally $latex($resp. prelocally$)$ in $latex\underline{S}^{p}$ $latex($resp. $latex\underline{H}^{p})$ to $latexM$ if $latexM^{n},M$ are in $latex\underline{S}^{p}$ (resp. $latex\underline{H}^{p}$) and if there exixts a sequence of stopping times $latexT_{k}$ increasing to $latex\infty$ a.s. such that
\[\lim_{n\rightarrow\infty}\parallel (M^{n}-M)^{T_{k}}\cdot I_{\{T_{k}>0\}}\parallel_{\underline{S}^{p}}=0\mbox{ (resp.}\lim_{n\rightarrow\infty}\parallel (M^{n}-M)^{T_{k}-}
\parallel_{\underline{H}^{p}}=0)\]
for each $latexk\geq 1$. $latex\sharp$

\begin{equation}{\label{prot511}}\tag{9}\mbox{}\end{equation}

Proposition \ref{prot511}. Let $latexJ,J^{n}\in {\bf D}$; $latexZ$ be a semimartingale with $latexZ_{0}=0$; and $latexF,F^{n}$ be functional Lipschitz with Lipschitz processes $latexK,K_{n}$ respectively. Let $latexX^{n}, X$ be solutions of
\begin{equation}{\label {proeq5*n1}}
X_{t}^{n}=J_{t}^{n}+\int_{0}^{t}F^{n}(X^{n})_{s-}dZ_{s}
\end{equation}
and
\begin{equation}{\label {proeq5*1}}
X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s},
\end{equation}
We assume

• \(J^{n}\) converges to $latexJ$ prelocally in $latex\underline{S}^{2}$;
• \(F^{n}(X)\) converges to $latexF(X)$ prelocally in $latex\underline{S}^{2}$, where $latexX$ is the solution of $latex(\ref{proeq5*1})$;
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s. for a non-random constant $latexa$.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ prelocally in $latex\underline{S}^{2}$ where $latexX^{n}$ is the solution of (\ref{proeq5*n1})  and $latexX$ is the solution of  (\ref{proeq5*1}) . $latex\sharp$

We can recast Proposition~\ref{prot511} in terms of convergence in ucp.

Corollary. Let $latexJ,J^{n}\in {\bf D}$; $latexZ$ be a semimartingale with $latexZ_{0}=0$; and $latexF,F^{n}$ be functional Lipschitz with Lipschitz processes $latexK,K_{n}$ respectively. Let $latexX^{n}, X$ be as in Proposition \ref{prot511}. We assume

• \(J^{n}\) converges to $latexJ$ in ucp;
• \(F^{n}(X)\) converges to $latexF(X)$ in ucp;
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s. for a non-random constant $latexa$.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ in ucp. $latex\sharp$

Proposition. Let $latexH^{n},H\in {\bf D}$. For $latexH^{n}$ to converge to $latexH$ in ucp it is necessary and sufficient that there exists a subsequence $latex\{n_{k}\}$ such that $latex\lim_{n_{k}\rightarrow\infty}H^{n_{k}}=H$ prelocally in $latex\underline{S}^{p}$. $latex\sharp$

One can weaken the hypothesis of Proposition~\ref{prot59} and still let the differentials vary provided the coefficients stay bounded, as the next result shows.

\begin{equation}{\label{prot513}}\tag{10}\mbox{}\end{equation}

Proposition \ref{prot513}. Let $latexJ,J^{n}\in {\bf D}$; $latexZ,Z^{n}$ be semimartingales with $latexZ_{0}=Z_{0}^{n}=0$; and $latexF,F^{n}$ be functional Lipschitz with Lipschitz processes $latexK,K_{n}$ respectively. Let $latexX^{n},X$ be solutions of $latex(\ref{proeq5*n})$ and $latex(\ref{proeq5*})$ respectively. We assume

• \(J^{n}\) converges to $latexJ$ prelocally in $latex\underline{S}^{2}$;
• \(F^{n}(X)\) converges to $latexF(X)$ prelocally in $latex\underline{S}^{2}$, and the coefficients $latexF^{n},F$ are all bounded by $latexc<\infty$;
• \(Z^{n}\) converges to $latexZ$ prelocally in $latex\underline{H}^{2}$;
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s. for a non-random constant $latexa$.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ prelocally in $latex\underline{S}^{2}$. $latex\sharp$

The assumptions of prelocal convergence are a bit awkward. This type of convergence leads to a topology on the space of semimartingales which is the natural topology for convergence of semimartingale differentials, just as ucp is the natural topology for processes related to stochastic integration. Before definging a topology on the space of semimartingales, let us recall that we can define a “distance” on $latex{\bf D}$ by setting, for $latexY,Z\in {\bf D}$,
\[r(Y)=\sum_{n>0}2^{-n}\cdot E\left [1\wedge\sup_{0\leq t\leq n}|Y_{t}|\right ]\mbox{ and }d(Y,Z)=r(Y-Z).\]
This distance is compatible with uniform convergence on compacts in probability.

Using stochastic integration we can define, for a semimartingale $latexX$,
\[\hat{r}(X)=\sup_{|H|\leq 1}r(H\bullet X)\]
where the supremum is taken over all predictable processes bounded by one; then the {\bf semimartingale topology} is defined by the distance $latex\hat{d}(X,Y)=\hat{r}(X-Y)$. The semimartingale topology can be shown to make the space of semimartingales a topological vector space which is complete.

Proposition. Let $latex1\leq p<\infty$, let $latexX^{n}$ be a sequence of semimartingales, and let $latexX$ be a semimartingale. Then, we have the following properties.

(i) If $latexX^{n}$ converges to $latexX$ in the semimartingale topology, then there exixts a subsequence which converges prelocally in $latex\underline{H}^{p}$;

(ii) If $latexX^{n}$ converges to $latexX$ prelocally in $latex\underline{H}^{p}$, then it converges to $latexX$ in the semimartingale topology. $latex\sharp$

We are now able once again to recast a result in terms of ucp convergence, Proposition \ref{prot513} has the following corollary.

Corollary. Let $latexJ,J^{n}\in {\bf D}$; $latexZ,Z^{n}$ be semimartingales with $latexZ_{0}=Z_{0}^{n}=0$; and $latexF,F^{n}$ be functional Lipschitz with Lipschitz processes $latexK,K_{n}$ respectively. Let $latexX^{n},X$ be solutions of $latex(\ref{proeq5*n})$ and $latex(\ref{proeq5*})$ respectively. We assume

• \(J^{n}\) converges to $latexJ$ in ucp;
• \(F^{n}(X)\) converges to $latexF(X)$ in ucp, where $latexX$ is the solution of $latex(\ref{proeq5*})$, and moreover all the coefficients $latexF^{n}$ are bounded by a random $latexc<\infty$;
• \(Z^{n}\) converges to $latexZ$ in the semimartingale topology;
• \(\max\{\sup_{n}K_{n},K\}\leq a<\infty\) a.s.

Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ in ucp. $latex\sharp$

The next result extends Proposition~\ref{prot59} and the preceding corollary by relaxing the hypotheses on convergence and especially the hypothesis that all the coefficients be bounded.

Proposition. Let $latexJ,J^{n}\in {\bf D}$; $latexZ,Z^{n}$ be semimartingales with $latexZ_{0}=Z_{0}^{n}=0$; and $latexF,F^{n}$ be functional Lipschitz with Lipschitz processes $latexK$, the same for all $latexn$. Let $latexX^{n},X$ be solutions respectively of
\begin{equation}{\label {proeq5n*2}}
X_{t}^{n}=J_{t}^{n}+\int_{0}^{t}F^{n}(X^{n})_{s-}dZ_{s}^{n}
\end{equation}
and
\begin{equation}{\label {proeq5*2}}
X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s}.
\end{equation}
We assume

• \(J^{n}\) converges to $latexJ$ in ucp;
• \(F^{n}(X)\) converges to $latexF(X)$ in ucp, where $latexX$ is the solution of $latex(\ref{proeq5*2})$, and moreover all the coefficients $latexF^{n}$ are bounded by a random $latexc<\infty$;
• \(Z^{n}\) converges to $latexZ$ in the semimartingale topology.

Then $latexX_{n}$ converges to $latexX$ in ucp. $latex\sharp$

Proposition. (Dominated Convergence). Let $latexp,q,r$ be given such that $latex1/p+1/q=1/r$, where $latex1<r<\infty$. Let $latexZ$ be a semimartingale in $latex\underline{H}^{q}$, and let $latexH^{n}\in
\underline{S}^{p}$ such that $latex|H^{n}|\leq Y\in\underline{S}^{p}$ for all $latexn\geq 1$. Suppose $latex\lim_{n\rightarrow\infty}H_{t-}^{n}(\omega )=0$ for all $latex(t,\omega )$. Then
\[\lim_{n\rightarrow\infty}\left |\!\left |\int H_{s-}^{n}dZ_{s}\right |\!\right |_{\underline{H}^{r}}=0. \sharp\]

Another important topic is how to approximate solutions by difference solutions. We let $latex\{\sigma_{n}\}$ be a sequence of random partitions tending to the identity. Recall that for a process $latexY$ and a random
partition
\[\sigma =\{0=T_{0}\leq T_{1}\leq\cdots\leq T_{k_{n}}\},\]
we define
\[Y^{\sigma}\equiv Y_{0}\cdot I_{\{0\}}+\sum_{k}Y_{T_{k}}\cdot I_{(T_{k},T_{k+1}]}.\]
Note that if $latexY$ is adapted, RCLL (i.e. $latexY\in {\bf D}$), then $latex\{Y_{s}^{\sigma}\}_{s\geq 0}$ is left-continuous with right limits and adapted. It is convenient to have a version of $latexY^{\sigma}\in {\bf D}$,
occasionally, so we define
\[Y^{\sigma +}=\sum_{k}Y_{T_{k}}\cdot I_{[T_{k},T_{k+1})}.\]

\begin{equation}{\label{prot516}}\tag{11}\mbox{}\end{equation}

Proposition \ref{prot516}. Let $latexJ\in\underline{S}^{2}$, let $latexF$ be process Lipschitz with Lipschitz process $latexK\leq a<\infty$ a.s. and $latexF(0)\in\underline{S}^{2}$. Let $latexZ$ be a semimartingale in $latex{\cal S}(1/(2\cdot\sqrt{8}\cdot a))$, and let $latexX(\sigma )$ be the solution of
\begin{equation}{\label {proeq5*3}}
X_{t}=J_{t}+\int_{0}^{t}F(X^{\sigma +})_{s}^{\sigma}dZ_{s}
\end{equation}
for a random partition $latex\sigma$. If $latex\{\sigma_{n}\}$ is a sequence of random partitions tending to the identity, then $latexX(\sigma_{n})$ tends to $latexX$ in $latex\underline{S}^{2}$, where $latexX$ is the solution of
(\ref{proeq5*}). $latex\sharp$

Corollary. Let $latexJ\in {\bf D}$; $latexF$ be process Lipschitz; $latexZ$ be a semimartingale; and let $latex\{\sigma_{n}\}$ be a sequence of random partitions tending to the identity. Then $latex\lim_{n\rightarrow\infty}X(\sigma_{n})=X$ in ucp, where $latexX(\sigma_{n})$ is the solution of $latex(\ref{proeq5*3})$ with $latex\sigma$ being replaced by $latex\sigma_{n}$ and $latexX$ is the solution of (\ref{proeq5*}). $latex\sharp$

Proposition \ref{prot516} and its corollary give us a way to approximate the solution of a general stochastic differential equation with finite differences. Indeed, let $latexX$ be the solution of
\[X_{t}=J_{t}+\int_{0}^{t}F(X)_{s-}dZ_{s}\]
where $latexZ$ is a semimartingale and $latexF$ is process Lipschitz. For each random partition
\[\sigma_{n}=\{0=T_{0}^{n}\leq T_{1}^{n}\leq\cdots\leq T_{k_{n}}^{n}\},\]
we see that the random variables $latexX(\sigma_{n})_{T_{k}^{n}}$ verify the relations (writing $latex\sigma$ for $latex\sigma_{n}$, $latexX$ for $latexX(\sigma_{n})$,
$T_{k}$ for $latexT_{k}^{n}$)
\begin{eqnarray*}
X_{T_{0}} & = & J_{0}\\
X_{T_{k+1}} & = & X_{T_{k}}+J_{T_{k+1}}-J_{T_{k}}+F(X^{\sigma +})_{T_{k}}
\cdot (Z_{T_{k+1}}-Z_{T_{k}}).
\end{eqnarray*}
Then the solution of the finite difference equation above converges to the solution of (\ref{proeq5*}) under the appropriate hypotheses.

Proposition. Let $latexZ$ be a semimartingale and let $latexX={\cal E}(Z)$, the stochastic exponential of $latexZ$. That is, $latexX$ is the solution of
\[X_{t}=1+\int_{0}^{t}X_{s-}dZ_{s}.\]
Let $latex\{\sigma_{n}\}$ be a sequence of random partitions tending to the identity. Let
\[X^{n}=\prod_{i=1}^{k_{n}-1}\left (1+(Z^{T_{i+1}^{n}}-Z^{T_{i}^{n}})\right ).\]
Then $latex\lim_{n\rightarrow\infty}X^{n}=X$ in ucp. $latex\sharp$

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

The Markov Nature of Solutions.

One of the original motivations for the development of the stochastic integral was to study continuous strong Markov process (that is, diffusions), as solutions of stochastic differential equations. Let $latexW=\{W_{t}\}_{t\geq 0}$ be a standard Brownian motion in $latex\mathbb{R}^{n}$. K. Ito studied systems of differential equations of the form
\[X_{t}=X_{0}+\int_{0}^{t}f(s,X_{s})dW_{s}+\int_{0}^{t}g(s,X_{s})ds,\]
and under appropriate hypotheses on the coefficients $latexf,g$, he showed that a unique continuous solution exists and that it is strong Markov.

Now we have semimartingale differentials, and it is therefore natural to replace $latexdW$ and $latexds$ with general semimartingales and to study any resulting Markovian nature of the solution. If we insist that the solution itself be Markov then the semimartingale differentials should have independent increments; but if we need only to relate the solution to a Markov process, then more general results are available. We begin the brief treatment of Markov processes with a naive definition. Assume as given a filtered probability space $latex(\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)$ satisfying the usual conditions.

Definition. A process $latex{\bf Z}$ with values in $latex\mathbb{R}^{d}$ and adapted to $latex\{{\cal F}_{t}\}_{t\geq 0}$ is a {\bf simple Markov process} with respect to $latex\{{\cal F}_{t}\}_{t\geq 0}$ if for each $latext\geq 0$ the $latex\sigma$-fileds $latex{\cal F}_{t}$ and $latex\sigma ({\bf Z}_{u};u\geq t)$ are conditionally independent given $latex{\bf Z}_{t}$. $latex\sharp$

Thus one can think of the Markov property as a weakening of the property of independent increments. It is easy to see that the simple Markov property is equivalent to the following: For $latexu\geq t$ and for every $latexf$ bounded, Borel measurable,
\begin{equation}{\label {proeq5*4}}
\mathbb{E}[f({\bf Z}_{u})|{\cal F}_{t}]=\mathbb{E}[f({\bf Z}_{u})|\sigma ({\bf Z}_{t})].
\end{equation}
One thinks of this as “the best prediction of the future given the past and the present is the same as the best prediction of the future given the present”.

Using the equivalent relation (\ref{proeq5*4}), one can define a ransition function for a Markov process as follows, for $latexs<t$ and $latexf$ bounded, Borel measurable
\[\mathbb{P}_{s,t}({\bf Z}_{s},f)=\mathbb{E}[f({\bf Z}_{t})|{\cal F}_{s}].\]
Note that i $latexf({\bf x})=I_{A}({\bf x})$, the indicator function of a set $latexA$, then the preceding equality reduces to
\[\mathbb{P}\left\{{\bf Z}_{t}\in A|{\cal F}_{s}\right\}=\mathbb{P}_{s,t}({\bf Z}_{s},I_{A}).\]
Identifying $latexI_{A}$ with $latexA$, we often write $latex\mathbb{P}_{s,t}(Z_{s},A)$ on the right side above. When we speak of a Markov process without specifying the filtration of $latex\sigma$-algebras $latex\{{\cal F}_{t}\}_{t\geq 0}$, we mean implicitly that $latex{\cal F}_{t}=\sigma ({\bf Z}_{s};s\leq t)$, the natural filtration generated by the process.

It often happens that the transition function satisfies the relationship $latex\mathbb{P}_{s,t}=\mathbb{P}_{t-s}$ for $latext\geq s$. In this case, we say that the Markov process is time homogeneous, and the transition functions are a semigroup of operators, known as the transition semigroup $\{\mathbb{P}_{t}\}_{t\geq 0}$. In the time homogeneous case, the Markov property becomes
\[\mathbb{P}\left\{{\bf Z}_{t+s}\in A|{\cal F}_{t}\right\}=\mathbb{P}_{s}({\bf Z}_{t},A).\]
A stronger requirement that is often satisfied is that the Markov property hold for stopping times.

Definition. A time homogeneous simple Markov process is strong Markov if for any stopping time $latexT$ with $latexP\{T<\infty\}=1$, $latexs\geq 0$,
\[P\left\{{\bf Z}_{T+s}\in A|{\cal F}_{T}\right\}=P_{s}({\bf Z}_{T},A)\]
or equivalently
\[E\mathbb{P}f({\bf Z}_{T+s})|{\cal F}_{T}]=\mathbb{P}_{s}({\bf Z}_{T},f),\]
for any bounded, Borel measurable function $latexf$. $latex\sharp$

The fact that we defined the strong Markov property only for time homogeneous processes is not much of a restriction, since if $latex{\bf X}$ is an $latex\mathbb{R}^{d}$-valued simple Markov process, then it is
easy to see that the process $latex{\bf Z}_{t}({\bf X}_{t},t)$ is an $latex\mathbb{R}^{d+1}$-valued time homgeneous simple Markov process. Examples of strong Markov processes are Brownian motion and the Poisson process. In order to allow arbitrary initial condition, we need (in general) a larger probability space than the one on which $latexZ$ is defined. We therefore define
\[\bar{\Omega}=\mathbb{R}^{n}\times\Omega,\bar{\cal F}_{t}^{0}={\cal B}\times {\cal F}_{t},\bar{P}^{{\bf y}}=\epsilon_{{\bf y}}\times P,\]

where $latex{\cal B}$ denotes the Borel sets of $latex\mathbb{R}^{n}$, and $latex\epsilon_{{\bf y}}$ denotes the Dirac point mass measure at $latex{\bf y}$. For a point $latex\bar{\omega}({\bf y},\omega )\in\bar{\Omega}$, we
further define
\begin{equation}{\label {proeq5*5}}
{\bf X}_{0}(\bar{\omega})={\bf y}.
\end{equation}
Finally let $latex\bar{cal F}_{t}=\cap_{u>t}\bar{\cal F}_{u}^{0}$. A random variable $latexZ$ defined on $latex\Omega$ is considered to be extended automatically to $latex\bar{\Omega}$ by the rule $latex{\bf Z}(\bar{\omega})={\bf Z}(\omega )$ when $latex\bar{\omega}=({\bf y}, \omega )$.

Proposition. Let $latexZ^{j}$ be semimartingales for $latex1\leq j\leq d$, $latex{\bf H}^{{\bf x}}$ a vector of adapted process in $latex{\bf D}$ for each $latex{\bf x}\in \mathbb{R}^{n}$, and suppose $latex({\bf x},t,\omega )\mapsto H_{t}^{{\bf x}}(\omega )$ is $latex{\cal B}(\mathbb{R}^{n})\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}$ measurable. Let $latexF_{j}^{i}$ be functional Lipschitz and for each $latex{\bf x}\in \mathbb{R}^{n}$, $latex{\bf X}^{{\bf x}}$ is the unique solution of
\[({\bf X}_{t}^{{\bf x}})^{i}=({\bf H}_{t}^{{\bf x}})^{i}+\sum_{j=1}^{d}\int_{0}^{t}F_{j}^{i}({\bf X}^{{|bf x}})_{s-}dZ_{s}^{j}.\]

There exists a version of $latex{\bf X}^{{\bf x}}$ such that $latex({\bf x},t,\omega ) \mapsto {\bf X}_{t}^{{\bf x}}(\omega )$ is $latex{\cal B}(\mathbb{R}^{n})\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}$ measurable, and for each $latex{\bf x}$, $latex{\bf X}_{t}^{{\bf x}}$ is a RCLL solution of the equation. $latex\sharp$

\begin{equation}{\label{prot532}}\tag{12}\mbox{}\end{equation}

Proposition \ref{prot532}. Let $latex{\bf Z}=(Z^{1},\cdots ,Z^{d})$ be a vector of independent L\'{e}vy processes with $latex{\bf Z}_{0}={\bf 0}$, and let $latex\{f_{j}^{i}\}$ for $latex1\leq j\leq d$ and $latex1\leq i\leq n$ be Lipschitz functions. Let $latex{\bf X}_{0}$ be as in $latex(\ref{proeq5*5})$ and let $latex{\bf X}$ be the solution of
\begin{equation}{\label {proeq5*6}}
X_{t}^{i}={\bf X}_{0}^{i}+\sum_{j=1}^{d}\int_{0}^{t}f_{j}^{i}(s-,X_{s-})dZ_{s}^{j}.
\end{equation}
Then $latex{\bf X}$ is a Markov process under each $latex\bar{P}^{{\bf y}}$ and $latex{\bf X}$ is strong Markov if the $latexf_{j}^{i}$ are autonomous. $latex\sharp$

If the differentials are not L\'{e}vy processes but only strong Markov processes which are semimartingales, then the solutions of equations such as (\ref{proeq5*6}) in Proposition \ref{prot532} need not be Markov
processes. One does, however, have the following result

Proposition. Let $latexZ$ be a strong Markov processes with values in $latex\mathbb{R}$ and $latexZ_{0}=0$ such that $latexZ$ is a semimartingale. Let $latexf$ and $latexg$ be Lipschitz functions. Let $latexX_{0}$ be as in Proposition \ref{prot532} nd $latexX$ be the solution of
\[X_{t}=X_{0}+\int_{0}^{t}f(s-,X_{s-})dZ_{s}+\int_{0}^{t}g(s,X_{s-})ds.\]
Then the vector process $latex(X,Z)$ is Markov under $latex\bar{P}^{y}$ for each $latexy\in \mathbb{R}$, and strong Markov if $latexf$ and $latexg$ are autonomous. $latex\sharp$

Traditionally the most important Markovian solutions of stochastic differential equations are diffusions. Suppose as given a space $latex(\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)$ satisfying the usual conditions.

Definition. An adapted process $latex{\bf X}$ with values in $latex\mathbb{R}^{n}$ is a diffusion if it has continuous sample paths and if it satisfies the strong Markov property. $latex\sharp$

A restatement of Proposition \ref{prot532} yields the following

\begin{equation}{\label{prot536}}\tag{13}\mbox{}\end{equation}

Proposition \ref{prot536}. Let $latex{\bf W}=(W^{1},\cdots ,W^{d})$ be a $latexd$-dimensional standard Brownian motion with $latex{\bf W}_{0}={\bf 0}$, and let $latex\{f_{j}^{i}\}$ for $latex1\leq j\leq d$ and $latex1\leq i\leq n$, $latexg^{i}$ be autonomous Lipschitz functions. Let $latex{\bf X}_{0}^{i}$ as in Proposition~\ref{prot532} and let $latex{\bf X}$ be the solution of
\begin{equation}{\label {proeq5*7}}
X_{t}^{i}=X_{0}^{i}+\sum_{j=1}^{d}\int_{0}^{t}f_{j}^{i}({\bf X}_{s})dW_{s}^{j}+\int_{0}^{t}g^{i}({\bf X}_{s})ds.
\end{equation}
Then $latex{\bf X}$ is a diffusion. $latex\sharp$

In equation (\ref{proeq5*7}) of Proposition \ref{prot536}, the coefficients $latexf_{j}^{i}$ are called the {\bf diffusion coefficients} and the coefficients $latexg^{i}$ are called the drift coefficients. Note that a diffusion need not be semimartingale. Indeed any deterministic continuous function with paths of unbounded variation is a diffusion which is not a semimartingale.

An intuitive notion of a diffusion is to imagine a pollen grain floating downstream in a river. The grain is subject to two forces: the current of the river (drift), and the aggregate bombardment of the grain by the surrounding water molecules (diffusion). The coefficient $latexf(t,x)$ then represents the sensitivity of the particle at time $latext$ and place $latexx$ to the diffusion forces. For example if part of the river water is warmer at certain times and places (due to sunlight or industrial effluents, for example), then $latexf$ might be larger. Analogously $latexg$ would be larger when the river was flowing faster due to a steeper incline. We give some examples of diffusions.

Example. The stochastic exponential $latex\exp\left (W_{t}-\frac{1}{2}t\right )$ is a diffusion where $latexf(t,x)=x$ and $latexg(t,x)=0$.

Example. Consider the simple system
\[V_{t}=V_{0}+\int_{0}^{t}\sigma dW_{s}+\int_{0}^{t}\alpha V_{s}ds\mbox{ and }X_{t}=X_{0}+\int_{0}^{t}V_{s}ds.\]
The process $latexX$ can be used as a model of Brownian motion alternative to Einstein’s; it is called the Ornstein-Uhlenbeck Brownian motion, or simply the Ornstein-Uhlenbeck process. Note that here the process $latexX$ has paths of finite variation and hence the process $latex\{V_{t}\}_{t\geq 0}$ is a true velocity process for $latexX$. Using integration by parts we can verify that
\[V_{t}=e^{\alpha t}\cdot\left (V_{0}+\int_{0}^{t}e^{-\alpha s}\sigma dW_{s}\right )\]
is an explicit solution for $latexV$. Indeed
\begin{align*}
e^{-\alpha t}\cdot V_{t} & =V_{0}+\int_{0}^{t}(-\alpha e^{-\alpha s})\cdot V_{s}ds+\int_{0}^{t}e^{-\alpha s}dV_{s}\\
& =V_{0}-\alpha\cdot\int_{0}^{t}e^{-\alpha s}\cdot V_{s}ds+\int_{0}^{t}e^{-\alpha s}\alpha\cdot V_{s}ds+\int_{0}^{t}e^{-\alpha s}\sigma dW_{s}\\
& =V_{0}+\int_{0}^{t}e^{-\alpha s}\sigma dW_{s},
\end{align*}
and we are done. Since $latexV_{0}$ and the Browian motion $latexW$ are independent (by construction), we see that when $latexV_{0}$ has a Gaussian distribution then $latexV$ is a Gaussian process. If $latex\alpha$ is negative and we take $latex\sigma^{2}=-1/2\alpha$, then $latexV$ is a stationary Gaussian process. $latex\sharp$

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

Flows of Stochastic Differential Equations: Continuity and Differentiability.

Consider a stochastic differential equation of the form
\[X_{t}=x+\int_{0}^{t}F(X)_{s-}dZ_{s}.\]
Obviously there is a dependence on the initial condition, and we can write the solution in the form $latexX(t,\omega ,x)$, or $latexX_{t}^{x}(\omega )$. The study of the flow of a stochastic differential equation is the study of the function $latex\phi :x\mapsto X(t,\omega ,x)$ which can be considered as mapping $latex\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ for $latex(t,\omega )$ fixed, or as mapping $latex\mathbb{R}^{n}\rightarrow {\cal D}^{n}$, where $latex{\cal D}^{n}$ denotes the sapce of RCLL functions from $latex\mathbb{R}_{+}$ to $latex\mathbb{R}^{n}$, equipped with the topology of uniform convergence on compacts. It is important to distinguish between $latex{\cal D}^{n}$ and $latex{\bf D}^{n}$: The former is a function space, and it is associated in the literature with weak convergence results; the latter is the space of stochastic processes with RCLL paths, and which are adapted to the underlying filtration. we will be interested in several properties of the flows: continuity, differentiability, injectivity, and when the flows are diffeomorphisms of $latex\mathbb{R}^{n}$. We begin with continuity. We consider a general system of equations of the form
\begin{equation}{\label {proeq5*8}}
{\bf X}_{t}^{{\bf x}}={\bf H}_{t}^{{\bf x}}+\int_{0}^{t}{\bf F}({\bf X}^{{\bf x}})_{s-}d{\bf Z}_{s}
\end{equation}
where $latex{\bf X}_{t}^{{\bf x}}$ and $latex{|bf H}_{t}^{{\bf x}}$ are $latex\mathbb{R}^{n}$ column vectors, $latex{\bf Z}$ is a column vector of $latexm$ semimartingales with $latex{\bf Z}_{0}={\bf 0}$, and $latex{\bf F}$ is an
$n\times m$ matrix with elements $latex\{F_{j}^{i}\}$ for $latex1\leq i\leq n$ and $latex1\leq j\leq m$. For $latex{\bf x}$ fixed, for each $latex{\bf y}$ we have that $latex\bar{{\bf X}}_{t}={\bf X}_{t}^{{\bf y}}-{\bf X}_{t}^{{\bf x}}$
is a solution of the equation
\begin{equation}{\label {proeq5*9}}
\bar{{\bf X}}_{t}={\bf H}_{t}^{{\bf y}}-{\bf H}_{t}^{{\bf x}}+\int_{0}^{t}\bar{{\bf F}}(\bar{{\bf X}})_{s-}d{\bf Z}_{s},
\end{equation}
where $latex\bar{{\bf F}}({\bf Y})={\bf F}({\bf X}^{{\bf x}}+{\bf Y})-{\bf F}({\bf X}^{{\bf x}})$.

Proposition. Let $latex{\bf H}^{{\bf x}}$ be processes in $latex{\bf D}^{n}$, and let $latex{\bf x}\mapsto {\bf H}^{{\bf x}}:\mathbb{R}^{n}\rightarrow {\bf D}^{n}$ be prelocally Lipschitz continuous in $latex\underline{S}^{p}$, some $latexp>n$. Let $latex{\bf F}$ be an $latexn\times m$ matrix of functional Lipschitz operators $latex\{F_{j}^{i}\}$ for $latex1\leq i\leq n$ and $latex1\leq j\leq m$. Then theer exists a function $latex{\bf X}(t,\omega ,{\bf x})$ on $latex\mathbb{R}_{+}\times\Omega\times \mathbb{R}^{n})$ such that

• For each $latex{\bf x}$, the process $latex{\bf X}_{t}^{{\bf x}}(\omega )= {\bf X}(t,\omega ,{\bf x})$ is a solution of $latex(\ref{proeq5*8})$;
• For almost all $latex\omega$, the flow $latex{\bf x}\mapsto {\bf X}(\cdot ,\omega ,{\bf x})$ from $latex\mathbb{R}^{n}$ into $latex{\cal D}^{n}$ is continuous in the topology of unform convergence on compacts. $latex\sharp$

We will need some hypotheses

• Hypothesis (H1). $latexZ^{j}$ are given semimartingales with $latexZ_{o}^{j}=0$ for $latex1\leq j\leq m$.
• Hypothesis (H2). $latexf_{j}^{i}:\mathbb{R}^{n}\rightarrow \mathbb{R}$ are given functions for $latex1\leq i\leq n$ and $latex1\leq j\leq m$, and $latex{\bf f}({\bf x})$ denotes the $latexn\times m$ matrix $latex(f_{j}^{i}({\bf x}))$.

We will study the system of equation
\begin{equation}{\label {proeq5*10}}
X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}f_{j}^{i}({\bf X}_{s-})dZ_{s}^{j}\mbox{ for }1\leq i\leq n
\end{equation}
which we also write
\begin{equation}{\label {proeq5*11}}
{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf f}({\bf X}_{s-})d{\bf Z}_{s}
\end{equation}
where it is understood that $latex{\bf X}_{t}$ and $latex{\bf x}$ are column vectors in $latex\mathbb{R}^{n}$, $latex{\bf f}({\bf X}_{s-})$ is an $latexn\times m$ matrix, and $latex{\bf Z}$ is a column vector of $latexm$ semimartingales.

Definition. A function $latexf:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is said to be locally Lipschitz if there exists an increasing sequence of open sets $latex\Gamma_{k}$ such that $latex\cup_{k}\Gamma_{k}=\mathbb{R}^{n}$ and $latexf$ is Lipschitz with a constant $latexK_{k}$ on each $latexGamma_{k}$. $latex\sharp$

For example, if $latexf$ has continuous first partial drivatives, then it is locally Lipschitz, while if its continuous first partials are bounded, then it is Lipschitz. If $latexf,g$ are both Lipschitz, then their product $latexfg$ is locally Lipschitz.

\begin{equation}{\label{prot538}}\mbox{}\end{equation}

Proposition \ref{prot538}. Let $latex{\bf Z}$ as in {\em (H1)} and let the functions $latex\{f_{j}^{i}\}$ in {\em (H2)} be locally Lipschitz. Then there exists a function $latex\zeta ({\bf x},\omega ):\mathbb{R}^{n}\times\Omega\rightarrow [0,\infty ]$ such that for each $latex{\bf x}$, $latex\zeta ({\bf x},\cdot )$ is a stopping time, and there exists a unique solution of $latex(\ref{proeq5*11})$ up to $latex\zeta
({\bf x},\cdot )$ with $latex\limsup_{t\rightarrow\zeta ({\bf x},\cdot )}\parallel {\bf X}_{t}\parallel =\infty$ a.s. on $latex\{\zeta <\infty\}$. Moreover $latex{\bf x}\mapsto\zeta ({\bf x},\omega )$ is lower semicontinuous,
strictly positive, and the flow of $latex{\bf X}$ is continuous on $latex[0,\zeta ({\bf x},\cdot ))$. $latex\sharp$

We comment that for each $latex{\bf x}$ fixed the stopping time $latexT(\omega )=\zeta ({\bf x},\omega )$ is called an explosion time. Thus Proposition \ref{prot538} assures the existence and uniqueness of a solution up to an explosion time; and at that time, the solution does indeed explode in the sense that $latex\bar{\lim}_{t\rightarrow T}\parallel {\bf X}_{t}\parallel =+\infty$ on $latex\{T<\infty\}$. Note however if the coefficients $latex\{f_{j}^{i}\}$ in (H2) are (globally) Lipschitz, then $latex\zeta =\infty$ a.s. for all $latex{\bf x}$.

We next turn the attention to the differentiability of the flows. Now we consider the system of $latexn+n^{2}$ equations (assuming that the coefficients $latex\{f_{j}^{i}\}$ are at least $latexC^{1}$)
\begin{equation}{\label {proeq5*12}}
\begin{array}{l}
{\displaystyle X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}f_{j}^{i}({\bf X}_{s-})dZ_{s}^{j}}\\
{\displaystyle D_{kt}^{i}=\delta_{k}^{i}+\sum_{j=1}^{m}\sum_{i=1}^{n}\int_{0}^{t}\frac{\partial f_{j}^{i}}{\partial x_{j}}({\bf X}_{s-})D_{ks-}^{j}dZ_{s}^{j}}
\end{array}
\end{equation}
for $latex1\leq i\leq n$, where $latexD$ denotes an $latexn\times m$ matrix valued process and $latex\delta_{k}^{i}=1$ if $latexi=k$ and $latex0$ otherwise (Kronecker’s delta). Note that in equations (\ref{proeq5*12}) if $latexX$ is already known, then the second system is linear in $latexD$. Also note that the coefficients for the system (\ref{proeq5*12}) are not globally Lipschitzian, but if the first partials of the $latex\{f_{j}^{i}\}$ are locally Lipschitzian, then so also are the coefficients of (\ref{proeq5*12}).

Proposition. Let $latex{\bf Z}$ be as in {\em (H1)} and let the functions $latex\{f_{j}^{i}\}$ in (H2) have locally Lipschitz first partial derivatives. Then for almost all $latex\omega$ there exists a function $latex{\bf X}(t,\omega ,{\bf x})$ which is continuously differentiable in the open set $latex\{{\bf x}:\zeta ({\bf x},\omega )>t\}$, where $latex\zeta$ is the explosion time. If $latex\{f_{j}^{i}\}$ are globally Lipschitz then $latex\zeta =\infty$. Let
\[D_{k}(t,\omega ,{\bf x})\equiv\frac{\partial}{\partial x_{k}}{\bf X}(t,\omega ,{\bf x}).\]
Then, for each $latex{\bf x}$, the process $latex({\bf X}(\cdot ,\omega ,{\bf x}), D(\cdot ,\omega ,{\bf x}))$ is identically RCLL, and it is the solution of equations $latex(\ref{proeq5*12})$ on $latex[0,\zeta ({\bf x},\cdot ))$. $latex\sharp$

\begin{equation}{\label{prot540}}\tag{14}\mbox{}\end{equation}

Proposition \ref{prot540} (Protter \cite{pro}). Let $latex{\bf Z}$ be as in {\em (H1)} and let the functions $latex\{f_{j}^{i}\}$ in (H2) have locally Lipschitz derivatives up to order $latexN$, for some $latexN$, $latex0\leq N\leq\infty$. Then there exists a solution $latex{\bf X}(t,\omega ,{\bf x})$ to
\[X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}f_{j}^{i}({\bf X}_{s-})dZ_{s}^{j}\mbox{ for }1\leq i\leq n,\]
which is $latexN$ times continuously differentiable in the open set $latex\{{\bf x}:\zeta ({\bf x},\omega )>t\}$, where $latex\zeta$ is the explosion time of the solution. If the coefficients $latex\{f_{j}^{i}\}$ are globally Lipschitz, then $latex\zeta =\infty$. $latex\sharp$

Note that the coefficients $latex\{f_{j}^{i}\}$ in Proposition \ref{prot540} are locally Lipschitz of order $latexN$ if, for example, they have $latexN+1$ continuous partial derivatives; that is, if $latexf_{j}^{i}\in C^{N+1}(\mathbb{R}^{n})$, for each $latexi$ and $latexj$, then $latex\{f_{j}^{i}\}$ are locally Lipschitz of order $latexN$.

\begin{equation}{\label{f}}\tag{F}\mbox{}\end{equation}

Flows as Diffeomorphisms: The Continuous Case.

Now we study a system of differential equations of the form
\begin{equation}{\label {proeq5*13}}
X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}F_{j}^{i}({\bf X})_{s-}dZ_{s}^{j}\mbox{ for }1\leq i\leq n,
\end{equation}
where the semimartingales $latexZ^{j}$ are assumed to have continuous paths with $latex{\bf Z}_{0}={\bf 0}$. The continuity assumption leads to pleasing results. The flows of an equation as (\ref{proeq5*13}) is considered to be an $latex\mathbb{R}^{n}$-valued function $latex\phi :\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ given by $latex\phi ({\bf x})={\bf X}(t,\omega ,{\bf x})$ for each $latex(t,\omega )$. We first
consider the possible injectivity of $latex\phi$.

Definition. The flow $latex\phi$ of equation $latex(\ref{proeq5*13})$ is said to be weakly injective if for each fixed $latex{\bf x},{\bf y}\in\mathbb{R}^{n}$ with $latex{\bf x}\neq {\bf y}$,
\[P\{\omega :\exists t, {\bf X}(t,\omega ,{\bf x})={\bf X}(t,\omega ,{\bf y})\}=0.\]
The flow $latex\phi$ of equation $latex(\ref{proeq5*13})$ is said to be strongly injective (or, simply, injective) if for almost all $latex\omega$ the function $latex\phi :{\bf x}\rightarrow {\bf X}(t,\omega ,{\bf x})$ is injective
for all $latext$. $latex\sharp$

Proposition. Let $latexZ^{j}$ be continuous semimartingales for $latex1\leq j\leq m$, $latex{\bf H}$ a vector of adapted RCLL processes, and $latex{\bf F}$ an $latexn\times m$ matrix of process Lipschitz operators. Then the flow of the solution of
\[{\bf X}_{t}={\bf x}+{\bf H}_{t}+\int_{0}^{t}{\bf F}({\bf X})_{s-}d{\bf Z}_{s}\]
is weakly injective. $latex\sharp$

Recall that if $latexZ$ is a continuous semimartingale, then $latexX_{0}\cdot {\cal E}(Z)$ denotes the (unique) solution of the equation
\[X_{t}=X_{0}+\int_{0}^{t}X_{s}dZ_{s},\]
and $latex{\cal E}(Z)_{t}=\exp\left (Z_{t}-\frac{1}{2}[Z,Z]_{t}\right )$. In particular, $latexP\{\inf_{s\leq t}{\cal E}(Z)_{s}>0\}=1$.

Proposition. Let $latexZ^{j}$ be continuous semimartingales for $latex1\leq j\leq m$, and $latex{\bf F}$ an $latexn\times m$ matrix of process Lipschitz operators. Then the flow of the solution of
\begin{equation}{\label {proeq5*14}}
{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf F}({\bf X})_{s-}d{\bf Z}_{s}
\end{equation}
is strongly injective on $latex\mathbb{R}^{n}$. $latex\sharp$

Proposition. Let $latexZ^{j}$ be continuous semimartingales for $latex1\leq j\leq m$, and let $latex{\bf F}$ be an $latexn\times m$ matrix pf process Lipschitz operators. Let $latex{\bf X}$ be the solution of $latex(\ref{proeq5*14})$. Then for each $latexN<\infty$ and almost all $latex\omega$
\[\lim_{\parallel {\bf x}\parallel\rightarrow\infty}\inf_{s\leq N}\parallel {\bf X}(s,\omega ,{\bf x})\parallel =\infty .\sharp\]

Proposition. Let $latexZ^{j}$ be continuous semimartingales for $latex1\leq j\leq m$, and $latex{\bf F}$ be an $latexn\times m$ matrix of process Lipschitz operators. Let $latex{\bf X}$ be the solution of (\ref{proeq5*14}). Let $latex\phi :\mathbb{R}^{n} \rightarrow\mathbb{R}^{n}$ be the flow $latex\phi ({\bf x})={\bf X}(t,\omega , {\bf x})$. Then for almost all $latex\omega$ one has that for all $latext$ the function $latex\phi$ is surjective and moreover it is a homeomorphism from $latex\mathbb{R}^{n}$ to $latex\mathbb{R}^{n}$. $latex\sharp$

We next turn the attention to determining when the flow is a diffeomorphism of $latex\mathbb{R}^{n}$. Recall that a diffeomorphism of $latex\mathbb{R}^{n}$ is a bijection which is $latexC^{\infty}$ and which has an inverse that is also $latexC^{\infty}$. For given $latexn$, let $latex{\bf Z}$ be an $latexn\times n$ matrix of given semimartingales. If $latex{\bf X}$ is a solution of
\[{\bf X}_{t}={\bf I}+\int_{0}^{t}{\bf X}_{s-}d{\bf Z}_{s},\]
where $latex{\bf X}$ is an $latexn\times n$ matrix of semimartingales and $latexI$ is the identity matrix, then $latex{\bf X}={\cal E}({\bf Z})$, the (matrix-valued) exponential of $latex{|bf Z}$. Since the space of $latexn\times n$ matrices is not commutative, it is also possible to consider right stochastic integrals, denoted
\[({\bf Z}:{\bf H})_{t}=\int_{0}^{t}(d{\bf Z}_{s}){\bf H}_{s},\]
where $latex{\bf Z}$ is an $latexn\times n$ matrix of semimartingales and $latex{\bf H}$ is an $latexn\times n$ matrix of (integrable) predictable processes. If $latex{\bf A}^{t}$ denotes matrix transpose, then
\[({\bf Z}:{\bf H})=({\bf H}^{t}\bullet {\bf Z}^{t})^{t},\]
and therefore right stochastic integrals can be defined in terms of stochastic inetgrals. Note that $latex\int {\bf Y}_{-}d{\bf Z}$ and $latex[{\bf Y},{\bf Z}]$ denote $latexn\times n$ matrix valued processes here.

\begin{equation}{\label{prot547}}\tag{15}\mbox{}\end{equation}

Proposition \ref{prot547}. Let $latex{\bf Y},{\bf Z}$ be given $latexn\times n$ matrices of semimartingales, $latex{\bf H}$ an $latexn\times n$ matrix of locally bounded predictable processes. Then
\begin{align*}
& {\bf Y}_{t}{\bf Z}_{t}-{\bf Y}_{0}{\bf Z}_{0}=\int_{0}^{t}{\bf Y}_{s-}d{\bf Z}_{s}+\int_{0}^{t}(d{\bf Y}_{s}){\bf Z}_{s-}+[{\bf Y},{\bf Z}]_{t}\\
& [{\bf H}\bullet {\bf Y},{\bf Z}]={\bf H}\bullet [{\bf Y},{\bf Z}]\\
& [{\bf Y},{\bf Z}:{\bf H}]=[{\bf Y},{\bf Z}]:{\bf H}
\end{align*}
Moreover if $latex{\bf F}$ is an $latexn\times n$ matrix of functional Lipschitz operators, then there exists a unique $latexn\times n$ matrix of $latex{\bf D}$-valued processes which is the solution of
\[{\bf X}_{t}={\bf I}+\int_{0}^{t}(d{\bf Z}_{s}){\bf F}({\bf X})_{s-}. \sharp\]

Proposition \ref{prot547} allows the definition of the right stochastic exponential

Definition. The right stochastic exponentia} of an $latexn\times n$ matrix of semimartingales $latex{\bf Z}$, denoted by $latex{\cal E}^{R}({\bf Z})$, is the (unique) matrix-valued solution of the equation
\[{\bf X}_{t}={\bf I}+\int_{0}^{t}(d{\bf Z}_{s}){\bf X}_{s-}. \sharp\]

\begin{equation}{\label{prot548}}\tag{16}\mbox{}\end{equation}

Proposition \ref{prot548}. Let $latex{\bf Z}$ be an $latexn\times n$ matrix of continuous semimartingales with $latex{\bf Z}_{0}={\bf 0}$. Then $latex{\cal E}({\bf Z})$ and $latex{\cal E}^{R} (-{\bf Z}+[{\bf Z},{\bf Z}])$ are inverse; that is, $latex{\cal E}({\bf Z})\cdot {\cal E}^{R}(-{\bf Z}+[{\bf Z},{\bf Z}])={\bf I}$. $latex\sharp$

Theorem.  Let $latex(Z^{1},\cdots ,Z^{m})$ be continuous semimartingales and let $latex\{f_{j}^{i}\}$ for $latex1\leq i\leq n$ and $latex1\leq j\leq m$ be functions mapping $latex\mathbb{R}^{n}$ to $latex\mathbb{R}$, with partial derivatives of all orders, and bounded first partials. Then the flow of the solution of
\[X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}f_{j}^{i}({\bf X}_{s})dZ_{s}^{j}\mbox{ for }1\leq i\leq n,\]

is a diffeomorphism from $latex\mathbb{R}^{n}$ to $latex\mathbb{R}^{n}$. $latex\sharp$

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

General Stochastic Exponentials and Linear Equations.

Let $latexZ$ be a given continuous semimartingale with $latexZ_{0}=0$ and let $latex{\cal E}(Z)_{t}$ denote the unique solution of the stochastic exponential equation
\[X_{t}=1+\int_{0}^{t}X_{s}dZ_{s}.\]
Then
\[X_{t}={\cal E}(Z)_{t}=\exp\left (Z_{t}-\frac{1}{2}[Z,Z]_{t}\right ).\]
It is of course unusual to have a closed form solution of a stochastic differential equation, and it is therefore especially nice to be able to give an explicit solution of the stochastic exponential equation when it also has an exogeneous driving term. That is, we want to consider equations of the form
\begin{equation}{\label {proeq5*15}}
X_{t}=H_{t}+\int_{0}^{t}X_{s-}dZ_{s},
\end{equation}
where $latexH\in {\bf D}$ (RCLL and adapted), and $latexZ$ is a continuous semimartingale. A unique solution of (\ref{proeq5*15}) exists by Proposition \ref{prot57}. It is written $latex{\cal E}_{H}(Z)$.

\begin{equation}{\label{prot552}}\mbox{}\end{equation}

Proposition \ref{prot552}. Let $latexH$ be a semimartingale and let $latexZ$ be a continuous semimartingale with $latexZ_{0}=0$. Then the solution $latex{\cal E}_{H}(Z)$ of equation (\ref{proeq5*15}) is given by
\[{\cal E}_{H}(Z)_{t}={\cal E}(Z)_{t}\cdot\left (H_{0}+\int_{0+}^{t}{\cal E}(Z)^{-1}_{s}d(H_{s}-[H,Z]_{s})\right ). \sharp\]

Since $latex{\cal E}(Z)^{-1}_{t}=1/{\cal E}(Z)_{t}$ appears in the formula for $latex{\cal E}_{H}(Z)$, it is worthwhile to note that (for $latexZ$ a continuous semimartingale)
\[d\left (\frac{1}{{\cal E}(Z)}\right )=\frac{dZ-d[Z,Z]}{{\cal E}(Z)}\]
and also
\[\frac{1}{{\cal E}(Z)}={\cal E}(-Z+[Z,Z]).\]
The next result generalizes Proposition \ref{prot552} to the case where $latexH$ is not necessarily a semimartingale.

Proposition. Let $latexH$ be RCLL, adapted $latex($i.e. $latexH\in {\bf D})$, and let $latexZ$ be a continuous semimartingale with $latexZ_{0}=0$. Let $latexX_{t}={\cal E}_{H}(Z)_{t}$ be the solution of
\[X_{t}=H_{t}+\int_{0}^{t}X_{s-}dZ_{s}.\]
Then $latexX_{t}={\cal E}(Z)_{t}$ is given by
\[X_{t}=H_{t}+{\cal E}(Z)_{t}\cdot\int_{0}^{t}{\cal E}(Z)_{s}^{-1}(H_{s-}dZ_{s}-H_{s-}d[Z,Z]_{s}).\]
\end{Pro}

Theorem. (Comparison Theorem).
Let $latex\{Z^{j}\}_{1\leq j\leq m}$ be continuous semimartingales with $latexZ_{0}^{j}=0$, and let $latexF_{j}$ be process Lipschitz. Let $latexA$ be a continuous, adapted process with increasing paths, strictly increasing at $latext=0$. Let $latexG$ and $latexH$ be process Lipschitz functionals such that $latexG(X)_{t-}>H(X)_{t-}$ for any continuous semimartingale $latexX$. Finally, let $latexX$ and $latexY$ be the unique solutions of
\begin{align*}
& X_{t}=x_{0}+\int_{0+}^{t}G(X)_{s-}dA_{s}+\int_{0}^{t}{\bf F}(X)_{s-}d{\bf Z}_{s}\\
& Y_{t}=y_{0}+\int_{0+}^{t}H(Y)_{s-}dA_{s}+\int_{0}^{t}{\bf F}(Y)_{s-}d{\bf Z}_{s}
\end{align*}
where $latexx_{0}\geq y_{0}$ and $latexF$ and $latexZ$ are written in vector notation. Then $latexP\{\exists t>0:X_{t}\leq Y_{t}\}=0$. $latex\sharp$

Consider the system of linear equations given by
\begin{equation}{\label {proeq5*16}}
{\bf X}_{t}={\bf H}_{t}+\sum_{j=1}^{m}\int_{0}^{t}{\bf A}_{s-}^{j}{\bf X}_{s-}dZ_{s}^{j}
\end{equation}
where $latex{\bf H}$ is a vector of $latexn$ semimartingales, $latex{\bf X}$ takes values in $latex\mathbb{R}^{n}$, and $latex{\bf A}^{j}$ is an $latexn\times n$ matrix of adapted, RCLL processes. The processes $latexZ^{j}$, $latex1\leq
j\leq m$, are given, continuous semimartingales which are zero at zero. Define the operators $latexF_{j}$ on $latex{\bf D}^{n}$ by
\[{\bf F}({\bf X})_{t}={\bf A}_{t}^{j}{\bf X}_{t}\]
where $latex{\bf A}^{j}$ is the $latexn\times n$ matrix specified above. The operators $latexF_{j}$ are essentially process Lipschitz.

Before examing equation (\ref{proeq5*16}), consider the simple system
\[U_{t}^{i,k}=\delta_{k}^{i}+\sum_{j=1}^{m}\int_{0}^{t}\sum_{l=1}^{n}(A_{i,l}^{j})_{s-}U_{s-}^{l,k}dZ_{s}^{j}\]
where $latex\delta_{k}^{i}=1$ if $latexi=k$ and $latex0$ otherwise. Letting $latex{\bf I}$ denote the $latexn\times n$ identity matrix and writing the preceding in matrix notation yields
\begin{equation}{\label {proeq5*17}}
{\bf U}_{t}={\bf I}+\sum_{j=1}^{m}{\bf A}_{s-}^{j}{\bf U}_{s-}dZ_{s}^{j},
\end{equation}
where $latex{\bf U}$ takes its values in the space of $latexn\times n$ matrices of adapted processes in $latex{\bf D}$.

Proposition. Let $latex{\bf A}^{j}$, $latex1\leq j\leq m$, be $latexn\times n$ matrices of RCLL, adpated processes, and let $latex{\bf U}$ be the solution of $latex(\ref{proeq5*17})$. Let $latex{\bf X}^{{\bf x}}_{t}$ be the solution of $latex(\ref{proeq5*16})$, where $latex{\bf H}_{t}={\bf x}$ for $latex{\bf x}\in \mathbb{R}^{n}$. Then $latex{\bf X}^{{\bf x}}_{t}={\bf U}_{t}{\bf x}$and for almost all $latex\omega$, for all $latext$ and $latex{\bf
x}$, the matrix $latex{\bf U}_{t}(\omega )$ is invertible. $latex\sharp$

Let $latex{\bf U}^{-1}$ denote the $latexn\times n$ matrix-valued process with continuous trajectories a.s. defined by $latex({\bf U}^{-1})_{t}(\omega )=({\bf U}_{t}(\omega ))^{-1}$. Let $latex[{\bf H},Z^{j}]$ denote the column vector of $latexn$ components, the $latexi$th one of which is $latex[H^{i},Z^{j}]$.

Proposition. Let $latex{\bf H}$ be a colimn vector of $latexn$ semimartingales, $latexZ^{j}$, $latex1\leq j\leq m$, be continuous semimartingales with $latexZ^{j}_{0}=0$, and let $latex{\bf A}^{j}$, $latex1\leq j\leq m$, be $latexn\times n$ matrices of processes in $latex{\bf D}$. Let $latex{\bf U}$ be the solution of equation $latex(\ref{proeq5*17})$. Then the solution $latex{\bf X}^{{\bf H}}$ of $latex(\ref{proeq5*16})$ is given by
\[{\bf X}^{{\bf H}}_{t}={\bf U}_{t}{\bf H}_{0}+{\bf U}_{t}\int_{0+}^{t}{\bf U}_{s}^{-1}\left (d{\bf H}_{s}-\sum_{j=1}^{m}{\bf A}_{s-}^{j}d[{\bf H},Z^{j}]_{s}\right ). \sharp\]

\begin{equation}{\label{h}}\tag{H}\mbox{}\end{equation}

Flows as Diffeomorphisms: The General Case.

Now we study the same equations
\begin{equation}{\label {proeq5*19}}
X_{t}^{i}=x^{i}+\sum_{j=1}^{m}\int_{0}^{t}f_{j}^{i}({|bf X})_{s-}dZ_{s}^{j}\mbox{ for }1\leq i\leq n,
\end{equation}
except that the semimartingales $latex\{Z^{j}\}_{1\leq j\leq m}$ are no longer assumed to be continuous. For simplicity we still assume that $latex{\bf Z}_{0}={\bf 0}$. In the general case, it is not always true
that the flows of solutions are diffeomorphisms of $latex\mathbb{R}^{n}$. The system of equations (\ref{proeq5*19}) may also be written as
\begin{equation}{\label {proeq5*20}}
{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf f}({\bf X}_{s-})d{\bf Z}_{s}
\end{equation}
where $latex{\bf X}_{t}$ and $latex{\bf x}$ are column vectors in $latex\mathbb{R}^{n}$, $latex{\bf f}({\bf X}_{s-})$ is an $latexn\times m$ matrix, and $latex{\bf Z}$ is a column vector of $latexm$ semimartingales.

Hypothesis (H3). $latex{\bf f}$ is $latexC^{\infty}$ and bounded derivatives of all orders.

Choose an $latex\epsilon >0$, the actual size of which is yet to be determined. For $latex\{Z^{j}\}_{1\leq j\leq m}$ we can find stopping times $latex0=T_{0}<T_{1}<T_{2}<\cdots$ tending to $latex\infty$ a.s. such that
\[Z^{j,k}=(Z^{j})^{T_{k}-}-(Z^{j})^{T_{k-1}}\]
have an $latex\underline{H}^{\infty}$ norm less than $latex\epsilon$ (ref. Proposition \ref{prot55}). Note that by Proposition \ref{ptot51}, $latex[Z^{j,k},Z^{j,k}]_{\infty}^{1/2}<\epsilon$ as well, hence the jumps of
each $latexZ^{j,k}$ are smaller than $latex\epsilon$. Therefore all of the “large” jumps of $latexZ^{j,k}$ occur only at the times $latex\{T_{k}\}$ for $latexk\geq 1$.

Let $latexX_{t}^{k}({\bf x})$ denote the solution of (\ref{proeq5*19}) driven by the semimartingales $latexZ^{j,k}$. Outside of the interval $latex(T_{k-1},T_{k})$ the solution is
\[X_{t}^{k}({\bf x})=\left\{\begin{array}{ll}
{\bf x} & \mbox{for $latext\leq T_{k-1}$}\\
X_{T_{k}-}^{k}({\bf x}) & \mbox{for $latext\geq T_{k}$}
\end{array}\right .\]
Next define the linkage operators
\[H^{j}({\bf x})={\bf x}+{\bf f}({\bf x})\Delta {\bf Z}_{T_{j}},\]
using vector and matrix notation.

Proposition. The flow $latex\phi :{\bf x}\rightarrow {\bf X}_{t}({\bf x},\omega )$ of the solution $latex{\bf X}$ of $latex(\ref{proeq5*20})$ is a diffeomorphism if the collections of functions $latex{\bf x}\mapsto X_{t}^{j}({\bf x},\omega )$ and $latex{\bf x}\mapsto H^{j}({\bf x},\omega )$ are diffeomorphisms. $latex\sharp$

Proposition. Let $latexZ^{j}$ be semimartingales, $latex1\leq j\leq m$, with $latexZ_{0}^{j}=0$, and let $latex{\bf F}$ be an $latexn\times m$ matrix of process Lipschitz operators with nonrandom Lipschitz constant $latexK$. Let $latexH^{i}\in {\bf D}$ for $latex1\leq i\leq n$ $latex($RCLL and adapted$)$. If $latex\sum_{j=1}^{m}\parallel Z^{j}\parallel_{\underline{H}^{\infty}}<\epsilon$ for $latex\epsilon >0$ sufficiently small, then the flow of the solution of
\[{\bf X}_{t}={\bf x}+{\bf H}_{t}+\int_{0}^{t}{\bf F}({\bf X})_{s-} d{\bf Z}_{s}\]
is weakly injective. $latex\sharp$

For given $latexn$, let $latex{\bf Z}$ be an $latexn\times n$ matrix of given semimartingales. Recall that $latex{\bf X}={\cal E}({\bf Z})$ denote the matrix-valued exponential of $latex{\bf Z}$, and that $latex{\cal E}^{R}({\bf Z})$ denotes the matrix-valued right stochastic exponential of $latex{\bf Z}$. Recall that in Proposition~\ref{prot548} we showed that if $latex{\bf Z}$ is an $latexn\times n$ matrix of continuous semimartingales with $latex{\bf Z}_{0}={\bf 0}$, then
\[{\cal E}({\bf Z}){\cal E}^{R}(-{\bf Z}+[{\bf Z},{\bf Z}])={\bf I}\mbox{ or equivalently }{\cal E}(-{\bf Z}+[{\bf Z},{\bf Z}]){\cal E}^{R}({\bf Z})={\bf I}.\]
The general case is more delicate.

Proposition. Let $latex{\bf Z}$ be an $latexn\times n$ matrix of semimartingales with $latex{\bf Z}_{0}={\bf 0}$. Suppose that
\[{\bf W}_{t}=-{\bf Z}_{t}+[{\bf Z},{\bf Z}]_{t}^{c}+\sum_{0<s\leq t}({\bf I}+\Delta {\bf Z}_{s})^{-1}(\Delta {\bf Z}_{s})^{2}\]
is a well-defined semimartingale. Then
\[{\cal E}({\bf W})_{t}{\cal E}({\bf Z})_{t}={\bf I}\]
for all $latext\geq 0$. $latex\sharp$

Corollary. Let $latex{\bf Z}$ be a square matrix of semimartingales. If $latex\parallel {\bf Z}\parallel_{\underline{H}^{\infty}}<\epsilon$ for $latex\epsilon >0$ sufficiently small, then $latex{\cal E}^{R}({\bf Z})_{t}$ is invertibale for all $latext\geq 0$. $latex\sharp$

Proposition. Let $latex\{Z^{j}\}_{1\leq j\leq m}$ be semimartingales with $latex{\bf Z}_{0}={\bf 0}$, and let $latex{\bf f}$ be a matrix of coefficients satisfying Hypotheses (H2) and  (H3). Let $latex{\bf X}$ be the unique solution of
\[{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf f}({\bf X}_{s-})d{\bf Z}_{s}.\]
The Jacobian matrix
\[D_{k}^{i}(t,\omega ,{\bf x})=\frac{\partial}{\partial x_{k}}X^{i}(t,\omega ,{\bf x})\]
is invertible for each $latext\geq 0$ provided $latex\parallel {\bf Z}\parallel_{\underline{H}^{\infty}}<\epsilon$ for sufficiently small $latex\epsilon >0$. $latex\sharp$

Before stating the principal result, we need to define two subsets of $latex\mathbb{R}^{m}$; recall that under Hypotheses (H1), (H2), and (H3), that $latex{\bf Z}$ is a given $latexm$-tuple of semimartingales and that $latex{\bf f}({\bf x})=(f_{j}^{i}({\bf x}))$ is an $latexn\times m$ matrix of $latexC^{\infty}$ functions. Let
\begin{align*}
{\cal D} & = \{{\bf z}\in \mathbb{R}^{m}:{\bf H}({\bf x})={\bf x}+{\bf f}({\bf x}){\bf z}\mbox{ is a diffeomorphism of }\mathbb{R}^{n}\}\\
{\cal I} & = \{{\bf z}\in \mathbb{R}^{m}:{\bf H}({\bf x})={\bf x}+{\bf f}({\bf x}){\bf z}\mbox{ is injective in }\mathbb{R}^{n}\}.
\end{align*}
Clearly $latex{\cal D}\subset {\cal I}$.

Theorem. Let $latex{\bf Z}$ and $latex{\bf f}$ be as given in Hypothese (H1), (H2), and (H3), and let $latex{\bf X}$ be the solution of
\[{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf f}({\bf X}_{s-})d{\bf Z}_{s}.\]
The flow of $latex{\bf X}$ is a.s. a diffeomorphism of $latex\mathbb{R}^{n}$ (resp. trajectories of $latex{\bf X}$ from different initial points a.s. never meet) for all $latext$ if and only if all the jumps of $latex{\bf Z}$ belong to $latex{\cal D}$ $latex($resp. all the jumps of $latex{\bf Z}$ belong to $latex{\bf I})$. $latex\sharp$

Corollary. Let $latex{\bf Z}$ and $latex{\bf f}$ be as given in Hypothese (H1), (H2), and (H3), and let $latex{\bf X}$ be the solution of
\[{\bf X}_{t}={\bf x}+\int_{0}^{t}{\bf f}({\bf X}_{s-})d{\bf Z}_{s}.\]
Then different trajectories of $latex{\bf X}$ can meet only at the jumps of $latex{\bf Z}$. $latex\sharp$

\begin{equation}{\label{i}}\tag{I}\mbox{}\end{equation}

Miscellaneous.

Stochastic Integrals.

A stochastic process $latexX=\{X_{t}\}_{t\geq 0}$ is called {\bf simple} if there is a partition $latex0=t_{0}<t_{1}<t_{2}<\cdots <t_{n}=T<\infty$ and uniformly bounded $latex{\cal F}_{t_{n}}$-measurable random variables $latex\xi_{k}$ ($|\xi_{k}|\leq C$ for all $latexk=0,1,\cdots ,n$ and $latex\omega$ for some $latexC$) and if $latexX_{t}(\omega )$ can be written in the form
\[X_{t}(\omega )=\xi_{0}\cdot 1_{\{0\}}(t)+\sum_{i=0}^{n}\xi_{i}(\omega )\cdot 1_{(t_{i},t_{i+1}]}(t)\mbox{ for }0\leq t\leq T, \omega\in\Omega .\]
Then if $latext_{k}\leq t<t_{k+1}$
\[I_{t}(X)\equiv\int_{0}^{t}XdW=\sum_{i=0}^{k-1}\xi_{i}(W_{t_{i+1}}-W_{t_{i}})+\xi_{k}(W_{t}-W_{t_{k}})=\sum_{i=0}^{n}\xi_{i}(W_{t\wedge t_{i+1}}-W_{t\wedge t_{i}}).\]
Note that by definition $latexI_{0}(X)=0$ $latexI\!\! P$-a.s..

Proposition. (Binham and Kiesel \cite{bin}) We have the following properties

(i) $latexI_{t}(aX+bY)=aI_{t}(X)+bI_{t}(Y)$.

(ii) $latexE[I_{t}(X)|{\cal F}_{s}]=I_{s}(X)$ $latexI\!\! P$-a.s. for $latex0\leq s\leq t<\infty$, hence $latexI_{t}(X)$ is a continuous martingale. $latex\sharp$

Proposition. (Binham and Kiesel \cite{bin}) We have the following properties.

(i) We have the Ito isometry
\[I\!\! E\left [(I_{t}(X))^{2}\right ]=I\!\! E\left [\int_{0}^{t}X^{2}_{u}du\right ].\]

(ii) We also have
\[I\!\! E\left [(I_{t}(X)-I_{s}(X))^{2}|{\cal F}_{s}\right ]=I\!\! E\left [\int_{s}^{t}X^{2}_{u}du\right ]\mbox{ $latexP$-a.s.}. \sharp\]

The Ito isometry above suggests that $latex\int_{0}^{t} XdW$ should be defined only for processes with
\begin{equation}{\label {bineq11}}
\int_{0}^{t} I\!\! E\left [X_{u}^{2}\right ]du<\infty\mbox{ for all }t.
\end{equation}
We then can transfer convergence on a suitable $latexL^{2}$-space of stochastic processes to a suitable $latexL^{2}$-space of martingales. This gives us an $latexL^{2}$-theory of stochastic integration. The suitable class of integrands is the class of $latex({\cal B}([0,\infty ))\times {\cal F})$-measurable, $latex{\cal F}_{t}$-adapted processes $latexX$ with equation (\ref{bineq11}) being satisfied. Each such $latexX$ may be approximated by a sequence of simple integrands $latexX_{n}$ so that the stochastic integral $latexI_{t}(X)=\int_{0}^{t} XdW$ may be defined as the limit of $latexI_{t}(X_{n})=\int_{0}^{t}X_{n}dW$. Suppose that $latex\{\mu_{t}\}$ is adapted and locally integrable, so $latex\int_{0}^{t}\mu_{u}du$ is defined as an ordinary integral, and $latex\{\sigma_{t}\}$ is adapted and measurable with $latex\int_{0}^{t}\mathbb{E}[\sigma^{2}_{u}]du<\infty$ for all $latext$, so $latex\int_{0}^{t}\sigma_{u}dW_{u}$ is defined as a stochastic integral. Then
\[X_{t}\equiv x_{0}+\int_{0}^{t}\mu_{u}dS+\int_{0}^{t}\sigma_{u}dW_{u}\]
defines a stochastic process $latexX$ with $latexX_{0}=x_{0}$. It is customary and convenient to express such an equation symbolically in differential form in terms of the stochastic differential equation
\begin{equation}{\label {bineq12}}
dX_{t}=\mu_{t}dt+\sigma_{t}dW_{t}\mbox{ with }X_{0}=x_{0}.
\end{equation}
Now suppose $latexf:I\!\! R\rightarrow I\!\! R$ is of class $latexC^{2}$. The question arises of giving a meaning to the stochastic differential $latexdf(X_{t})$ of the process $latexf(X_{t})$ and finding it.

\begin{equation}{\label{binp65}}\mbox{}\end{equation}

Theorem \ref{binp65}. (Ito’s Lemma). Suppose that $latexX$ has stochastic differential given by $latex($\ref{bineq12}$)$.

(i) If $latexf\in C^{2}$ then $latexf(X)$ has stochastic differential
\[df(X_{t})=f'(X_{t})dX_{t}+\frac{1}{2}f”(X_{t})d\langle X\rangle_{t},\]
or writing out the integrals
\[f(X_{t})=f(x_{0})+\int_{0}^{t}f'(X_{u})dX_{u}+\frac{1}{2}\int_{0}^{t}f”(X_{u})d\langle X\rangle_{u}.\]

(ii) If $latexf\in C^{2}$ then $latexf=f(t,X_{t})$ has the stochastic differential
\[df=\left (f_{t}+bf_{x}+\frac{1}{2}\sigma^{2}f_{xx}\right )dt+\sigma f_{x}dW.\]
That is, writing $latexf_{0}=f(0,X_{0})$, the initial value of $latexf$,
\[f=f_{0}+\int_{0}^{t}\left (f_{t}+bf_{x}+\frac{1}{2}\sigma^{2}f_{xx}\right )dt+\int_{0}^{t}\sigma f_{x}dW. \sharp\]

If $latexf(t,x_{1},x_{2})=x_{1}x_{2}$ and $latexX$ and $latexY$ are suitable semimartingales, we obtain the product rule
\begin{equation}{\label {bineq48}}
X_{t}Y_{t}=\int_{0}^{t}X_{u}dY_{u}+\int_{0}^{t}Y_{u}dX_{u}+\langle X,Y\rangle_{t}.
\end{equation}
If
\[X_{t}=X_{0}+\int_{0}^{t}\mu_{u}^{(1)}du+\int_{0}^{t}\sigma_{u}^{(1)}dW_{u}\mbox{ and }
Y_{t}=Y_{0}+\int_{0}^{t}\mu_{u}^{(2)}du+\int_{0}^{t}\sigma_{u}^{(2)}dW_{u},\]
then
\begin{equation}{\label {museqb17}}
\langle X,Y\rangle_{t}=\int_{0}^{t}\sigma_{u}^{(1)}\cdot\sigma_{u}^{(2)}du
\end{equation}

Suppose we wish to model the time evolution of a stock price $latexS_{t}$. Consider how $latexS$ will change in some small time interval from the present time $latext$ to a time $latext+dt$ in the near future. Writing $latexdS_{t}$ for the change $latexS_{t+dt}-S_{t}$ in $latexS$, the return on $latexS$ in this interval is $latexdS_{t}/S_{t}$. It is economically reasonable to expect this return to decompose into two components, a systematic part and a random part. The systematic part could plausibly be modelled by $latex\mu dt$, where $latex\mu$ is some parameter representing the mean rate of return of the stock. The random part could plausibly be modelled by $latex\sigma dW_{t}$, where $latexdW_{t}$ represents the noise term driving the stock price dynamics, and $latex\sigma$ is a second parameter describing how much effect this noise — how much the stock price fluctuates. Thus $latex\sigma$ governs how volatile the price is, and is called the {\bf volatility} of the stock. Putting this together, we have the stochastic differential equation
\[dS_{t}=S_{t}(\mu dt+\sigma W_{t})\mbox{ with }S_{0}>0.\]
The differential equation above has the unique solution
\[S_{t}=S_{0}\exp\left [\left (\mu-\frac{1}{2}\sigma^{2}\right )t+\sigma W_{t}\right ].\]
In particular,
\[\log S_{t}=\log S_{0}+\left (\mu-\frac{1}{2}\sigma^{2}\right )t+\sigmaW_{t}\]
has a normal distribution. Thus $latexS_{t}$ itself has a lognormal distribution.

\begin{equation}{\label{mustb13}}\mbox{}\end{equation}

Proposition \ref{mustb13}. (Musiela and Rutkowski \cite{mus}). For any random variable $latexX\in L^{2}(\Omega ,{\cal F}_{T},I\!\! P)$, there exists a unique predictable process $latex\gamma$ from the class
${\cal F}_{I\!\! P}(W)$ such that
\[I\!\! E_{I\!\! P}\left [\int_{0}^{T}\gamma_{u}^{2}du\right ]<\infty\]
and the following equality is valid
\[X=I\!\! E_{I\!\! P}[X]+\int_{0}^{T}\gamma_{u}dW_{u}. \sharp\]

It can be deduced from Proposition~\ref{mustb13} that any local martingale on the filtered probability space $latex(\Omega ,I\!\! F^{W},I\!\! P)$ admits a modification with continuous sample paths, where $latexI\!\! F^{W}$ is the standard augmentation of the natural filtration $latex\sigma (\{W_{u}:u\leq t\})$ of the Brownian motion $latexW$.

The Snell Envelope.

Definition. If $latex\{Z_{n}\}_{n=0}^{N}$ is a sequence adapted to a filtration $latex\{{\cal F}_{n}\}$, the sequence $latex\{U_{n}\}_{n=0}^{N}$ defined by
\[\left\{\begin{array}{l}
U_{N}\equiv Z_{N},\\
U_{n}\equiv\max\{Z_{n},E[U_{n+1}|{\cal F}_{n}]\}\mbox{ for }n\leq N-1
\end{array}\right .\]
is called the {\bf Snell envelope} of $latex\{Z_{n}\}$. $latex\sharp$

Theorem (Bingham and Kiesel \cite{bin}). The Snell envelope $latex\{U_{n}\}$ of $latex\{Z_{n}\}$ is a supermartingale, and is the smallest supermartingale dominating $latex\{Z_{n}\}$; i.e.,
$U_{n}\geq Z_{n}$ for all $latexn$. $latex\sharp$

Proposition (Bingham and Kiesel \cite{bin}). $latexT_{0}\equiv\inf\{n\geq 0:U_{n}=Z_{n}\}$ is a stopping time, and the stopped sequence $latex\{U_{n\wedge T_{0}}\}$ is a martingale. $latex\sharp$

We write $latex{\cal T}_{n,N}$ for the set of stopping times taking values in $latex\{n,n+1,\cdots ,N\}$. We next see that the Snell envelope solves the optimal stopping problem.

Proposition (Bingham and Kiesel \cite{bin}). $latexT_{0}$ solves the optimal stopping problem for $latex\{Z_{n}\}$
\[U_{0}=E[Z_{T_{0}}|{\cal F}_{0}]=\sup\{E[Z_{T}|{\cal F}_{0}]:T\in {\cal T}_{0,N}\}. \sharp\]

The same argument, starting at time $latexn$ rather than time $latex0$, gives

Corollary (Bingham and Kiesel \cite{bin}). If $latexT_{n}\equiv\inf\{j\geq n:U_{j}=Z_{j}\}$, then
\[U_{n}=E[Z_{T_{n}}|{\cal F}_{n}]=\sup\{E[Z_{T}|{\cal F}_{n}]:T\in {\cal T}_{n,N}\}. \sharp\]

As we are attempting to maximize the payoff by stopping $latex\{Z_{n}\}$ at the most advantageous time, the Corollary shows that $latexT_{n}$ gives the best stopping time that is realistic: it maximizes the expected payoff given only information currently available. We thus call $latexT_{0}$ or $latexT_{n}$ the optimal stopping time for the problem.

Stochastic Calculus for Black-Scholes Models.

Consider first independent $latexN(0,1)$ random variables $latexZ_{1},Z_{2},\cdots ,Z_{n}$ on a probability space $latex(\Omega ,{\cal F},I\!\! P)$. Given a vector $\boldsymbol{\gamma}=(\gamma_{1},\cdots ,\gamma_{n})$, consider a new probability measure $latex\bar{I\!\! P}$ on $latex(\Omega ,{\cal F})$ defined by
\[\bar{I\!\! P}(d\omega )=\exp\left [\sum_{i=1}^{n}\gamma_{i}Z_{i}(\omega )-\frac{1}{2}\sum_{i=1}^{n}\gamma_{i}^{2}\right ]I\!\! P(d\omega ).\]
$\bar{I\!\! P}$ is also equivalent to $latexI\!\! P$ (has the same null sets). We also have
\begin{align*}
\bar{I\!\! P}\{Z_{1}\in dz_{1},\cdots ,Z_{n}\in dz_{n}\} & = exp\left [\sum_{i=1}^{n}\gamma_{i}z_{i}-\frac{1}{2}\sum_{i=1}^{n}\gamma_{i}^{2}\right ]I\!\! P\{Z_{1}\in dz_{1},\cdots ,Z_{n}\in dz_{n}\}\\
& =(2\pi )^{-n/2}\exp\left [\sum_{i=1}^{n}z_{i}-\frac{1}{2}\sum_{i=1}^{n}\gamma_{i}^{2}-\frac{1}{2}\sum_{i=1}^{n}z_{i}^{2}\right ]dz_{1}\cdots dz_{n}\\
& =(2\pi )^{-n/2}\exp\left [-\frac{1}{2}\sum_{i=1}^{n}(z_{i}-\gamma_{i})^{2}\right ]dz_{1}\cdots dz_{n}.
\end{align*}
This says that if the $latexZ_{i}$ are independent $latexN(0,1)$ under $latexI\!\! P$, they are independent $latexN(\gamma_{i},1)$ under $latex\bar{I\!\! P}$. Thus the effect of the change of measure $latexI\!\! P$ to $latex\bar{I\!\! P}$, from the original measure $latexI\!\! P$ to the equivalent measure $latex\bar{I\!\! P}$, is to change the mean, from $latex{\bf 0}=(0,\cdots ,0)$ to $\boldsymbol{\gamma}= (\gamma_{1},\cdots ,\gamma_{n})$.

This result extends to infinitely many dimensional, i.e., from random vectors to stochastic processes. Let $latex{\bf W}=(W_{1},\cdots ,W_{d})$ be a $latexd$-domensional Brownian motion defined on a filtered probability space $latex(\Omega ,{\cal F},I\!\! P,I\!\! F)$ with the filtration $latexI\!\! F$ satisfying the usual condition. Let $latex\{\boldsymbol{\gamma}_{t}:0\leq t\leq T\}$ be a measurable, adapted $latexd$-dimensional process with $latex\int_{0}^{T} (\gamma_{t}^{(i)})^{2}dt<\infty$ a.s. for $latexi=1,\cdots ,d$, and define the process $latex\{L_{t}:0\leq t\leq T\}$ by
\[L_{t}\equiv\exp\left [-\int_{0}^{t}\boldsymbol{\gamma}_{u}\cdot d{\bf W}_{u}-\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right ].\]
Then $latexL$ is continuous and is a local martingale. Given sufficient integrability on the process $\boldsymbol{\gamma}$, $latexL$ will in fact be a (continuous) martingale. For this, Novikov’s condition suffices
\[I\!\! E_{I\!\! P}\left [\exp\left (\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\right ]<\infty.\]

\begin{equation}{\label{binp66}}\tag{14}\mbox{}\end{equation}

Theorem \ref{binp66} (Bingham and Kiesek \cite{bin})(Girsanov’s Theorem). Let $\boldsymbol{\gamma}$ be as above and satisfy the Novikov’s condition. Let $latexL$ be the corresponding continuous martingale. Define the processes $latex\bar{W}_{i}$ for $latexi=1,\cdots ,d$ by
\[\bar{W}_{i}(t)\equiv W_{i}(t)+\int_{0}^{t}\gamma_{i}(u)du\mbox{ for }0\leq t\leq T\mbox{ and }i=1,\cdots ,d.\]
or
\[d\bar{W}_{i}(t)=dW_{i}(t)+ \gamma_{i}(t)dt\mbox{ for }0\leq t\leq T\mbox{ and }i=1,\cdots ,d.\]
Then under the equivalent probabilty measure $latex\bar{I\!\! P}$ defined on $latex(\Omega ,{\cal F}_{T})$ with Radon-Nikodym derivative
\[\frac{d\bar{I\!\! P}}{dI\!\! P}=L_{T},\]
the process $latex\bar{{\bf W}}=(\bar{W}_{1},\cdots ,\bar{W}_{d})$ is $latexd$-dimensional Brownian motion. $latex\sharp$

\begin{equation}{\label{mustb21}}\tag{15}\mbox{}\end{equation}

Theorem \ref{mustb21}  (Musiela and Rutkowski \cite{mus}) (Girsanov’s Theorem). Let $latex{\bf W}$ be a standard $latexd$-dimensional Brownian motion on a filtered probability space $latex(\Omega ,{\cal F},I\!\! P)$. Suppose that $\boldsymbol{\gamma}$ is an adapted $latexI\!\! R^{d}$-valued process such that
\[I\!\! E_{I\!\! P}\left [\exp\left (\int_{0}^{T}\boldsymbol{\gamma}_{u}\cdot d{\bf W}_{u}-\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\right ]=1.\]
Define a probability measure $latex\bar{I\!\! P}$ on $latex(\Omega ,{\cal F}_{T})$ equivalent to $latexI\!\! P$ by means of the Radon-Nikodym derivative (or density)
\[\frac{d\bar{I\!\! P}}{dI\!\! P}=\exp\left (\int_{0}^{T}\boldsymbol{\gamma}_{u}\cdot d{\bf W}_{u}-\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right ),
\mbox{ $latexI\!\! P$-a.s.}\]
Then the process $latex\bar{{\bf W}}$, which is given by the formula
\[\bar{{\bf W}}_{t}={\bf W}_{t}-\int_{0}^{t}\boldsymbol{\gamma}_{u}du\mbox{ for all }t\in [0,T]\]
follows a standard $latexd$-dimensional Brownian motion on the space$(\Omega ,{\cal F},I\!\! P)$. $latex\sharp$

\begin{equation}{\label{binp68}}\tag{16}\mbox{}\end{equation}

Theorem \ref{binp68}. (Representation Theorem). Let $latexM=\{M_{t}\}_{t\geq 0}$ be a RCLL local martingale with respect to the Brownian filtration $latex\{{\cal F}_{t}\}$. Then
\[M_{t}=M_{0}+\int_{0}^{t}H_{u}dW_{u},t\geq 0\]
with $latexH=\{H_{t}\}_{t\geq 0}$ a progressively measurable process such that $latex\int_{0}^{t}H_{u}^{2}du<\infty$, $latext\geq 0$ with probability one. That is, all Brownian local martingales may be represented as stochastic integrals with respect to Brownian motion (and as such are continuous). $latex\sharp$

\begin{equation}{\label{binp79}}\tag{17}\mbox{}\end{equation}

Theorem \ref{binp79}. (Feynman-Kac Formula). The solution $latexF=F(t,x)$ to the partial differential equation
\[F_{t}+\mu F_{x}+\frac{1}{2}\sigma^{2}F_{xx}=0\]
with final condition $latexF(T,x)=h(x)$ has the stochastic representation
\[F(t,x)=E[h(X_{T})|X_{t}=x],\]
where $latexX$ satisfies the stochastic differential equation
\[dX_{u}=\mu (u,X_{u})du+\sigma (u,X_{u})dW_{u}, t\leq u\leq T\]
with initial condition $latexX_{t}=x$. $latex\sharp$