Stochastic Calculus For Brownian Motions

August Stephan Sedlacek (1868-1936) was an Austrian painter.

The topics are

Convergence in L^{2} \ref{a}
Approach II (Convergence in L^{2}
Approach III (Convergence in Probability)
Approach IV (Convergence in Probability)

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

Convergence in L^{2}

We are going to consider the approach of convergence in $L^{2}$. This approach follows from Kloeden and Platen \cite{klo}. We shall consider the Ito integral of a stochastic process $X$ over the unit time interval $0\leq t\leq 1$, denoting it by $I(X)$, where
\[I(X)(\omega )=\int_{0}^{1}X_{s}(\omega )dW_{s}(\omega ).\]
For a nonrandom step function $X_{t}(\omega )=c_{j}$ on $t_{j}\leq t< t_{j+1}$ for $j=1,2,\cdots ,n$, where $0=t_{1}<t_{2}<\cdots <t_{n+1}=1$, we should obviously take, with probability one,
\begin{equation}{\label {kloeq319}}\tag{1}
I(X)(\omega )=\sum_{j=1}^{n}c_{j}\cdot (W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega )).
\end{equation}
This is a random variable with zero mean since it is the sum of random variables with zero mean. For random step functions appropriate measurability conditions must be imposed to ensure the non-anticipativeness of the integral. To be specific, suppose that $\{{\cal F}_{t}\}_{t\geq 0}$ is a filtration such that $W_{t}$ is ${\cal F}_{t}$-measurable for each $t\geq 0$. Then we consider a random step function $X_{t}(\omega )=\xi_{j}(\omega )$ on $t_{j}\leq t<t_{j+1}$ for $j=1,2,\cdots ,n$, where $0=t_{1}<t_{2}<\cdots <t_{n+1}=1$ and $\xi_{j}$ is ${\cal F}_{t_{j}}$-measurable, that is observable by events that can be detected at or before time $t_{j}$. We shall also assume that each $\xi_{j}$ is mean-square integrable over $\Omega$, hence $\mathbb{E}[\xi_{j}^{2}]<\infty$ for $j=1,2,\cdots ,n$. Since $\mathbb{E}[W_{t_{j+1}}-W_{t_{j}}|{\cal F}_{t_{j}}]=0$ a.s., it follows that the product $\xi_{j}\cdot (W_{t_{j+1}}-W_{t_{j}})$, which is ${\cal F}_{t_{j+1}}$-measurable and integrable, has expectation
\[\mathbb{E}[\xi_{j}\cdot (W_{t_{j+1}}-W_{t_{j}})]=\mathbb{E}[\mathbb{E}[\xi_{j}\cdot (W_{t_{j+1}}-
W_{t_{j}})|{\cal F}_{t_{j}}]]=\mathbb{E}[\xi_{j}\cdot \mathbb{E}[(W_{t_{j+1}}-W_{t_{j}})|{\cal F}_{t_{j}}]]=0\]
for each $j=1,2,\cdots ,n$. Analogously to (\ref{kloeq319}) we define the integral $I(X)$ by
\begin{equation}{\label {kloeq3110}}\tag{2}
I(X)(\omega )=\sum_{j=1}^{n}\xi_{j}(\omega )\cdot (W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega ))\mbox{ a.s.}
\end{equation}
Since the $j$th term in this sum is ${\cal F}_{t_{j+1}}$-measurable and hence ${\cal F}_{1}$-measurable (note that we consider the unit time interval $0\leq t\leq 1$), it follows that $I(X)$ is ${\cal F}_{1}$-measurable. In addition, $I(X)$ is integrable over $\Omega$ and has zero mean; in fact, it is mean-square integrable with
\begin{equation}{\label {kloeq3111}}\tag{3}
\mathbb{E}\left [I^{2}(X)\right ]=\sum_{j=1}^{n}\mathbb{E}\left [\xi_{j}^{2}\cdot
\mathbb{E}\left .\left [|W_{t_{j+1}}-W_{t_{j}}|^{2}\right |{\cal F}_{t_{j}}\right ]
\right ]=\sum_{j=1}^{n}\mathbb{E}[\xi_{j}^{2}]\cdot (t_{j+1}-t_{j})
\end{equation}
on account of the mean-square property of the increments $W_{t_{j+1}}-W_{t_{j}}$ for $j=1,2,\cdots ,n$. Finally, from (\ref{kloeq3110}) we obviously have
\begin{equation}{\label {kloeq3112}}\tag{4}
I(\alpha X_{1}+\beta X_{2})=\alpha I(X_{1})+\beta I(X_{2})\mbox{ a.s.}
\end{equation}
for any $\alpha ,\beta\in \mathbb{R}$ and any random step functions $X_{1},X_{2}$ satisfying the above properties, that is, the integration operator $I$ is linear in the integrand.

For a general integrand $X:[0,1]\times\Omega\rightarrow \mathbb{R}$, we shall define $I(X)$ as the limit of integrals $I(X^{(m)})$ of random step functions $X^{(m)}$ converging to $X$. To do this, we
need to specify conditions on $X$ and determine an appropriate mode of convergence for which such an approximating sequence of step functions and limit exist. For the moment, we shall assume that $X$
is continuous in $t$ for all $\omega\in\Omega$ and that for each $0\leq t\leq 1$ the random variable $X_{t}$ is ${\cal F}_{t}$-measurable and mean-square integrable with $\mathbb{E}[X_{t}^{2}]$ continuous in $t$. Then we form a partition $\pi_{n}=\left\{0=t_{1}^{(n)}<t_{2}^{(n)}<\cdots<t_{N_{n}+1}^{(n)}=1\right\}$ with
\[\parallel\pi_{n}\parallel =\max_{1\leq j\leq N_{n}}\left\{t_{j+1}^{(n)}-t_{j}^{(n)}\right\}\rightarrow 0\mbox{ as }n\rightarrow\infty\]
and define a step function $X^{(m)}$ by $X^{(m)}_{t}(\omega )=X_{\tau_{j}^{(n)}}(\omega )$ on $t_{j}^{(n)}\leq t<t_{j+1}^{(n)}$ for some choice of $\tau_{j}^{(n)}$ satisfying $t_{j}^{(n)}\leq\tau_{j}^{(n)}<t_{j+1}^{(n)}$, where $j=1,2,\cdots ,N_{n}$. For a choice $t_{j}^{(n)}<\tau_{j}^{(n)}<t_{j+1}^{(n)}$, the random variable $X^{(m)}_{t}$ need not be ${\cal F}_{t}$-measurable for each $t_{j}^{(n)}\leq t<t_{j+1}^{(n)}$ nor independent of the increment $W_{t_{j+1}}-W_{t_{j}}$, that is the step function $X^{(m)}$ may depend on the future events. To avoid this we must take $\tau_{j}^{(n)}=t_{j}^{(n)}$ for all $j=1,2,\cdots ,N_{n}$ and $n=1,2,\cdots$. We shall assume that we have done this and that the step functions $X^{(m)}$ converge to the integrand $X$ in an appropriate mode of convergence. The problem is to characterize the limit of the finite sums
\begin{equation}{\label {kloeq3113}}\tag{5}
I(X^{(m)})(\omega )=\sum_{j=1}^{n}X^{(m)}_{t_{j}^{(n)}}(\omega )\cdot\left (W_{t_{j+1}^{(n)}}(\omega )-W_{t_{j}^{(n)}}(\omega )\right )
\end{equation}
with respect to an appropriate mode of convergence. The Brownian motion $W$ has well-behaved mean-square properties, in particular $\mathbb{E}[(W_{t}-W_{s})^{2}]=t-s$. Moreover for the step functions $X^{(m)}$ the
equality (\ref{kloeq3111}) gives
\begin{equation}{\label {kloeq3114}}\tag{6}
\mathbb{E}\left [I^{2}(X^{(m)})\right ]=\sum_{j=1}^{n}\mathbb{E}[X^{2}_{t_{j}^{(n)}}]\cdot (t_{j+1}-t_{j})
\end{equation}
and this converges to the Riemann integral $\int_{0}^{1}\mathbb{E}[X^{2}_{s}]ds$ for $n\rightarrow\infty$ since $\mathbb{E}[X^{2}_{t}]$ has been assumed to be continuous in $t$. Together these suggest that we should use mean-square convergence, and this is exactly what Ito did. If we assume that $\mathbb{E}[|X^{(m)}_{t}-X_{t}|^{2}]\rightarrow 0$ as $m\rightarrow\infty$ (i.e. $X^{(m)}\rightarrow X$ in $L^{2}$) for all $t\in [0,1]$, then it is not hard to show that the mean-square limit of the $I(X^{(m)})$ exists and is unique a.s. (This will be seen later). We shall denote it by $I(X)$ and call it the Ito integral of $X$ (i.e. $I(X^{(m)})\rightarrow I(X)$ in $L^{2}$, or the integration operator $I$ is continuous). $I(X)$ turns out to be an ${\cal F}_{1}$-measurable random variable which is mean-square integrable on $\Omega$ with $\mathbb{E}[I(X)]=0$ and
\[\mathbb{E}\left [I^{2}(X)\right ]=\int_{0}^{1}\mathbb{E}\left [X^{2}_{s}\right ]ds.\]
In addition, $I(X)$ inherits the linearity property (\ref{kloeq3112}) from sums (\ref{kloeq3113}). Later in the discussions we shall drop the requirement that $X$ has continuous sample paths, where the Ito integral is defined as above and has the same properties.

The Ito integral is defined similarly on any bounded interval $[t_{0},t]$, resulting in a random variable
\begin{equation}{\label {kloeq3116}}\tag{7}
Y_{t}(\omega )=\int_{t_{0}}^{t}X_{s}(\omega )dW_{s}(\omega )
\end{equation}
which is ${\cal F}_{t}$-measurable and mean-square integrable with zero mean and
\[\mathbb{E}[Y_{t}^{2}]=\int_{t_{0}}^{t}\mathbb{E}[X^{2}_{s}]ds.\]
From the independence of nonoverlapping increments of a Brownian motion in the step function sum (\ref{kloeq3113}), and in their mean-square limit, we have
\[\mathbb{E}[Y_{t_{2}}-Y_{t_{1}}|{\cal F}_{t_{1}}]=0\mbox{ a.s.}\]
for any $t_{0}\leq t_{1}\leq t_{2}$; hence $\{Y_{t}\}_{t\geq 0}$ is a martingale.

The Ito integral also has the peculiar property that
\begin{equation}{\label {kloeq3117}}\tag{8}
\int_{0}^{t} W_{s}(\omega )dW_{s}(\omega )=\frac{1}{2}W_{s}^{2}(\omega )-\frac{1}{2}t\mbox{ a.s.}
\end{equation}
in contrast to
\[\int_{0}^{t}w(s)dw(s)=\frac{1}{2}w^{2}(t)\]
from classical calculus for a differentiable function $w(t)$ with $w(0)=0$. The equality (\ref{kloeq3117}) follows from the algebraic rearrangement
\[\sum_{j=1}^{n}W_{t_{j}}\cdot (W_{t_{j+1}}-W_{t_{j}})=\frac{1}{2}W_{t}^{2}-\frac{1}{2}\sum_{j=1}^{n}(W_{t_{j+1}}-W_{t_{j}})^{2}\]
for any $0=t_{1}<t_{2}<\cdots <t_{n+1}=1$ and the fact that the mean-square limit of the sum of squares on the right is equal to $t$.

We could write the equality (\ref{kloeq3117}) symbolically in terms of differentials as $\mathbb{E}[(dW_{t})^{2}]=dt$, which has an interesting consequence in the following stochastic counterpart of the chain rule. For each $t\geq t_{0}$ we define a stochastic process $Z_{t}$ by $Z_{t}(\omega )=U(t,Y_{t}(\omega ))$, where $U(t,x)$ has continuous second partial derivatives and $Y_{t}$ is given by (\ref{kloeq3116}) or, equivalently, by the stochastic differential $dY_{t}=XdW_{t}$. If $Y_{t}$ were continuously differentiable, the chain rule of classical calculus would give as the differential for $Z_{t}$
\begin{equation}{\label {kloeq3118}}\tag{9}
dZ_{t}=\frac{\partial U}{\partial t}(t,Y_{t})dt+\frac{\partial U}{\partial x}(t,Y_{t})dY_{t}.
\end{equation}
This follows from the first term in the Taylor expansion for $U$ in $\Delta Z_{t}=U(t+\Delta t,Y_{t}+\Delta Y_{t})-U(t,Y_{t})$, with the second and higher order terms in $\Delta t$ and $\Delta Y_{t}$ being discarded. In this case the only first order partial derivatives of $U$ remain. In contrast, when $Y_{t}$ is given by (\ref{kloeq3116}) we need to take into account that $(dY_{t})^{2}=X^{2}(dW_{t})^{2}$ and hence $\mathbb{E}[(dY_{t})^{2}]=\mathbb{E}[X^{2}]dt$, giving us a first order “$dt$” term coming from the second order part of the Taylor expansion for $U$. To be specific, we have
\begin{align*}
\Delta Z_{t} & =U(t+\Delta t,Y_{t}+\Delta Y_{t})-U(t,Y_{t})\\
& =\left (\frac{\partial U}{\partial t}\Delta t+\frac{\partial U}{\partial x}\Delta Y_{t}\right )+\frac{1}{2}\left (\frac{\partial^{2}U}{\partial t^{2}}(\Delta t)^{2}+2\frac{\partial^{2}U}{\partial t\partial x}
\Delta t\Delta Y_{t}+\frac{\partial^{2}U}{\partial x^{2}}(\Delta Y_{t})^{2}\right )+\cdots
\end{align*}
Thus as we shall see later, we obtain
\begin{equation}{\label {kloeq3119}}\tag{10}
dZ_{t}=\left (\frac{\partial U}{\partial t}(t,Y_{t})+\frac{1}{2}X^{2}\frac{\partial^{2}U}{\partial x^{2}}(t,Y_{t})\right )dt+\frac{\partial U}{\partial x}(t,Y_{t})dY_{t}
\end{equation}
with equality interpreted in the mean-square sense. This is a stochastic chain rule and is known as the Ito formula. It contains an additional term not present in the usual chain rule (\ref{kloeq3118}), and this gives rise to the extra term in integrals like (\ref{kloeq3117}). For example, with $Y_{t}=W_{t}$ and $Z_{t}=Y_{t}^{2}$, so $X=1$ and $U(t,x)=x^{2}$, we have
\[dZ_{t}=d(Y_{t}^{2})=dt+2Y_{t}dY_{t}\mbox{ or }W_{t}dW_{t}=\frac{1}{2}d(W_{t}^{2})-\frac{1}{2}dt.\]
Hence in integral form
\[\int_{0}^{t}W_{s}dW_{s}=\frac{1}{2}\int_{0}^{t}d(W_{s}^{2})-\frac{1}{2}\int_{0}^{t}ds=\frac{1}{2}W_{t}^{2}-\frac{1}{2}t,\]
since $W_{0}=1$ a.s.

It is usual to express the Ito formula in terms of the differentials $dt$ and $dW_{t}$. This is easily done in (\ref{kloeq3119}) since $dY_{t}=XdW_{t}$. In the general case, a stochastic differential can also include a “$dt$” term, that is, it has the form
\[dX_{t}(\omega )=\mu_{t}(\omega )dt+\sigma_{t}(\omega )dW_{t}(\omega ).\]
Then the Ito formula has the form
\begin{equation}{\label {kloeq3120}}\tag{11}
dY_{t}=\left (\frac{\partial U}{\partial t}+\mu_{t}\cdot\frac{\partial U}
{\partial x}+\frac{1}{2}\sigma_{t}^{2}\cdot\frac{\partial^{2}U}{\partial x^{2}}\right )dt+\sigma_{t}\cdot\frac{\partial U}{\partial x}dW_{t}
\end{equation}
where $Y_{t}=U(t,X_{t})$ and the partial derivatives of $U$ are evaluated at $(t,X_{t})$. When $U$ is linear in $x$ we have $\partial^{2}U/\partial x^{2}=0$ and the Ito formula (\ref{kloeq3120}) reduces to the usual chain rule (\ref{kloeq3118}).

In the sequel, we shall consider the Ito stochastic integral and its properties from a more thorough mathematical perspective, at the same time extending the definition to a wider class of integrands that mentioned above. For this we suppose that we have a probability space $(\Omega ,{\cal F},P)$, a Brownian motion $\{W_{t}\}_{t\geq 0}$ and an increasing family $\{{\cal F}_{t}\}_{t\geq 0}$ of sub-$\sigma$-fields of ${\cal F}$ such that $W_{t}$ is ${\cal F}_{t}$-measurable with
\[\mathbb{E}[W_{t}|{\cal F}_{0}]=0\mbox{ and }\mathbb{E}[W_{t}-W_{s}|{\cal F}_{s}]=0\mbox{ a.s.}\]
for all $0\leq s\leq t$. Here the $\sigma$-field ${\cal F}_{t}$ may be thought of as a collection of events that are detectable prior to or at time $t$, so that the ${\cal F}_{t}$-measurability of $Z_{t}$ for a stochastic process $Z$ indicates its non-anticipativeness with respect to the Brownian motion $W$.

For $0<t<\infty$, we define a class ${\cal L}_{t}^{2}$ of stochastic process $X:[0,t]\times\Omega\rightarrow \mathbb{R}$ satisfying the following conditions

$X$ is jointly ${\cal B}\times {\cal F}$-measurable;
${\displaystyle \int_{0}^{t}\mathbb{E}[X^{2}_{s}]ds<\infty}$;
$\mathbb{E}[X^{2}_{s}]<\infty$ for each $0\leq s\leq t$;
$X_{s}$ is ${\cal F}_{s}$-measurable for each $0\leq s\leq t$.

In addition we consider two functions in ${\cal L}_{t}^{2}$ to be identical if they are equal for all $(s,\omega )$ except possibly on a subset of $\lambda\times P$-measure zero. Then with the norm
\begin{equation}{\label {kloeq325}}\tag{12}
\parallel X\parallel_{2,t}=\sqrt{\int_{0}^{t}\mathbb{E}[X^{2}_{s}]ds}
\end{equation}
${\cal L}_{t}^{2}$ is a complete normed linear space, that is a Banach space, provided we identify functions which differ only on sets of measure zero (in fact, it is a Hilbert space).

We remark that conditions (i)-(iv) are stronger than $X\in L^{2}([0,t]\times\Omega ,{\cal B}\times {\cal F},\lambda\times P)$ which, by Fubini’s Theorem guarantees (iii) only for almost all $\lambda$-measure
$s\in [0,t]$, rather than for all $s\in [0,t]$, as will be required below. For any partition $0=t_{1}<t_{2}<\cdots <t_{n+1}=t$ and any mean-square integrable ${\cal F}_{t_{j}}$-measurable random variables $\xi_{j}$ for $j=1,2,\cdots ,n$, we define a step function $X\in {\cal L}_{t}^{2}$ by $X_{s}(\omega )=\xi_{j}(\omega )$ a.s. for $t_{j}\leq s<t_{j+1}$ and $j=1,2,\cdots ,n$. Here the integral in (ii) takes the form (also ref. (\ref{kloeq3114}))
\[\int_{0}^{t}\mathbb{E}[X^{2}_{s}]ds=\sum_{j=1}^{n}\mathbb{E}[\xi^{2}_{j}]\cdot (t_{j+1}-t_{j}).\]
We denote by ${\cal S}_{t}^{2}$ the subset of all step functions in ${\cal L}_{t}^{2}$. Then we can approximate any function in ${\cal L}_{t}^{2}$ by step functions in ${\cal S}_{t}^{2}$ to any desired degree of accuracy in the norm (\ref{kloeq325}). To be specific we have the following result.

\begin{equation}{\label{klol321}}\tag{13}\mbox{}\end{equation}

Proposition \ref{klol321}. ${\cal S}_{t}^{2}$ is dense in $({\cal L}_{t}^{2},\parallel\cdot\parallel_{2,t})$. $\sharp$

Now let $X$ be a step function in ${\cal S}_{t}^{2}$ corresponding to a partition $0=t_{1}<t_{2}<\cdots <t_{n+1}=t$ and random variables $\xi_{1},\xi_{2},\cdots ,\xi_{n}$. We define the Ito stochastic integral for this $X$ over the interval $[0,t]$ by
\begin{equation}{\label {kloeq327}}\tag{14}
I(X)(\omega )=\sum_{j=1}^{n}\xi_{j}(\omega )\cdot (W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega ))\mbox{ a.s. }
\end{equation}
Since $\xi_{j}$ is ${\cal F}_{t_{j}}$-measurable and $W_{t_{j+1}}-W_{t_{j}}$ is ${\cal F}_{t_{j+1}}$-measurable where ${\cal F}_{t_{j}}\subseteq {\cal F}_{t_{j+1}}$, their product $\xi_{j}\cdot (W_{t_{j+1}}-W_{t_{j}})$ is ${\cal F}_{t_{j+1}}$-measurable for $j=1,2,\cdots ,n$; hence $I(X)$ is ${\cal F}_{t}$-measurable. In addition, each product is integrable over $\Omega$, which follows from the Cauchy-Schwartz inequality and the fact that each term is mean-square integrable; hence $I(X)$ is integrable. In fact
\[\mathbb{E}[I(X)]=\sum_{j=1}^{n}\mathbb{E}\left [\xi_{j}\cdot (W_{t_{j+1}}-W_{t_{j}})\right ]=\sum_{j=1}^{n}\mathbb{E}[\xi_{j}]\cdot \mathbb{E}[W_{t_{j+1}}-W_{t_{j}}|{\cal F}_{t_{j}}]=0\]
since $\mathbb{E}[W_{t_{j+1}}-W_{t_{j}}|{\cal F}_{t_{j}}]=0$. Also $\xi_{j}$ and $\xi_{i}\xi_{j}\cdot (W_{t_{i+1}}-W_{t_{i}})$ are ${\cal F}_{t_{j}}$-measurable for any $i<j$. Thus
\begin{align*}
\mathbb{E}[I^{2}(X)] & =\sum_{j=1}^{n}\mathbb{E}\left [\xi_{j}^{2}\cdot (W_{t_{j+1}}-W_{t_{j}})^{2}\right ]+2\sum_{j=1}^{n}\sum_{i=j+1}^{n}\mathbb{E}\left [\xi_{i}\xi_{j}
\cdot (W_{t_{j+1}}-W_{t_{j}})(W_{t_{i+1}}-W_{t_{i}})\right ]\\
& =\sum_{j=1}^{n}\mathbb{E}[\xi_{j}^{2}]\cdot \mathbb{E}\left .\left [(W_{t_{j+1}}-W_{t_{j}})^{2}\right |{\cal F}_{t_{j}}\right ]\\
& +2\sum_{j=1}^{n}\sum_{i=j+1}^{n}\mathbb{E}\left [\xi_{i}\xi_{j}\cdot (W_{t_{j+1}}-W_{t_{j}})\right ]\cdot\mathbb{E}\left .\left [W_{t_{i+1}}-W_{t_{i}}\right |{\cal F}_{t_{j}}\right ]\\
& =\sum_{j=1}^{n}\mathbb{E}[\xi_{j}^{2}]\cdot (t_{j+1}-t_{j})\\
& =\int_{0}^{t}\mathbb{E}[X^{2}_{s}]ds,
\end{align*}
where we have used $\mathbb{E}[W_{t_{j+1}}-W_{t_{j}}|{\cal F}_{t_{j}}]=0$, $\mathbb{E}[(W_{t_{j+1}}-W_{t_{j}})^{2}|{\cal F}_{t_{j}}]=t_{j+1}-t_{j}$ and the definition of the Lebesgue or Riemann integral for the nonrandom step function $\mathbb{E}[X^{2}_{s}]$. Finally we note that for $X^{(1)},X^{(2)}\in {\cal S}_{t}^{2}$ and $\alpha ,\beta\in\mathbb{R}$ we have $\alpha X^{(1)}+\beta X^{(2)}\in {\cal S}_{t}^{2}$, with the combined step points of $X^{(1)}$ and $X^{(2)}$, so by algebraic rearrangement we obtain
\[I(\alpha X^{(1)}+\beta X^{(2)})=\alpha I(X^{(1)})+\beta I(X^{(2)})\mbox{ a.s.}\]
Collecting these results we thus proved

\begin{equation}{\label{klol322}}\tag{15}\mbox{}\end{equation}

Proposition \ref{klol322}. For any $X^{(1)},X^{(2)}\in {\cal S}_{t}^{2}$ and $\alpha ,\beta\in \mathbb{R}$, the Ito integral (\ref{kloeq327}) satisfies the following properties:

(i) $I(X)$ is ${\cal F}_{t}$-measurable;

(ii) $\mathbb{E}[I(X)]=0$;

(iii) ${\displaystyle \mathbb{E}\left [I^{2}(X)\right ]=\int_{0}^{t}\mathbb{E}[X^{2}_{s}]ds}$;

(iv) $I(\alpha X^{(1)}+\beta X^{(2)})=\alpha I(X^{(1)})+\beta I(X^{(2)})$ a.s. $\sharp$.

For an arbitrary stochastic process $X\in {\cal L}_{t}^{2}$, Proposition \ref{klol321} provides us with a sequence of step functions $X^{(n)}\in {\cal S}_{t}^{2}$ for which
\begin{equation}{\label {kloeq*1}}\tag{16}
\int_{0}^{t}\mathbb{E}\left [\left |X^{(n)}_{s}-X_{s}\right |^{2}\right ]ds\rightarrow 0\mbox{ as }n\rightarrow\infty .
\end{equation}
The integrals $I(X^{(n)})$ are well-defined by (\ref{kloeq327}) and, since $I(X^{(n)})-I(X^{(n+m)})=I(X^{(n)}-X^{(n+m)})$ and using (iii) in Proposition \ref{klol322}, they satisfy
\[\mathbb{E}\left [\left |I(X^{(n)})-I(X^{(n+m)})\right |^{2}\right ]=\mathbb{E}\left [\left |
I(X^{(n)}-X^{(n+m)})\right |^{2}\right ]=\int_{0}^{t}\mathbb{E}\left [\left |X^{(n)}_{s}-X^{(n+m)}_{s}\right |^{2}\right ]ds,\]
which by the inequality $(a+b)^{2}\leq 2(a^{2}+b^{2})$ gives
\begin{equation}{\label {kloeq3212}}\tag{17}
\begin{array}{lll}
{\displaystyle \mathbb{E}\left [\left |I(X^{(n)})-I(X^{(n+m)})\right |^{2}\right ]}
& = & {\displaystyle \int_{0}^{t}\mathbb{E}\left [\left |
(X^{(n)}_{s}-X_{s})+(X_{s}-X^{(n+m)}_{s})\right |^{2}\right ]ds}\\
& \leq & {\displaystyle 2\int_{0}^{t}\mathbb{E}\left [\left |
X^{(n)}_{s}-X_{s}\right |^{2}\right ]ds+2\int_{0}^{t}\mathbb{E}\left [\left |
X_{s}-X^{(n+m)}_{s}\right |^{2}\right ]ds}
\end{array}
\end{equation}
From (\ref{kloeq*1}), we see that $\mathbb{E}\left [\left |I(X^{(n)})-I(X^{(n+m)})\right |^{2}\right ]\rightarrow 0$. This says that $I(X^{(n)})$ is a Cauchy sequence in the Banach space (complete space) $L^{2}(\Omega ,{\cal F},P)$, and so there exists a unique, a.s., random variable $I$ in $L^{2}(\Omega ,{\cal F},\mathbb{P})$ such that $\mathbb{E}\left [|I(X^{(n)})-I|^{2}\right ]\rightarrow 0$ as $n\rightarrow\infty$. This $I$ is ${\cal F}_{t}$-measurable since it is the limit of ${\cal F}_{t}$-measurable random variables. Moreover we obtain the same limit $I$ for any choice of step functions converging to $X$ in ${\cal L}_{t}^{2}$. To see this let $\{\tilde{X}^{(n)}\}$ be another sequence of step functions converging to $X$ and suppose that $I(\tilde{X}^{(n)})$ converges to $\tilde{I}$. Then
\[\mathbb{E}\left [|I-\tilde{I}|^{2}\right ]\leq 2\mathbb{E}\left [|I-I(X^{(n)})|^{2}\right ]+2\mathbb{E}\left [|\tilde{I}-I(X^{(n)})|^{2}\right ]\]
using the inequality $(a+b)^{2}\leq 2(a^{2}+b^{2})$ again and
\[\mathbb{E}\left [|\tilde{I}-I(X^{(n)})|^{2}\right ]\leq 2\mathbb{E}\left [|\tilde{I}-I(\tilde{X}^{(n)})|^{2}\right ]+2\mathbb{E}\left [|I(X^{(n)})-I(\tilde{X}^{(n)})|^{2}\right ]\]
Using (\ref{kloeq3212}), we have
\[\mathbb{E}\left [|I(X^{(n)})-I(\tilde{X}^{(n)})|^{2}\right ]\leq 2\int_{0}^{t}\mathbb{E}\left [\left |X^{(n)}_{s}-X_{s}\right |^{2}\right ]+2\int_{0}^{t}\mathbb{E}\left [\left |\tilde{X}^{(n)}_{s}-X_{s}\right |^{2}\right ].\]
Taking limit, we obtain $\mathbb{E}\left [|I-\tilde{I}|^{2}\right ]=0$, and hence $I=\tilde{I}$ a.s. (this is incorrect unless $E$ can be replaced by $\int_{A}$ for all $A\subseteq\Omega$).

We define the Ito stochastic integral $I(X)$ of a stochastic process $X\in {\cal L}_{t}^{2}$ to be the common mean-square limit of sequences of the sums (\ref{kloeq327}) for any sequence of step functions in ${\cal S}_{t}^{2}$ converging to $X$ in the norm (\ref{kloeq325}). It obviously inherits the properties listed in Proposition~\ref{klol322} for the step functions, we have

\begin{equation}{\label{klot323}}\tag{18}\mbox{}\end{equation}

Proposition \ref{klot323}. The Ito stochastic integral $I(X)$ satisfies properties (i)–(iv) in Proposition \ref{klol322} for functions $X\in {\cal L}_{t}^{2}$. $\sharp$

From the identity $4ab=(a+b)^{2}-(a-b)^{2}$, the linearity of Lebesgue and Ito integrals and property (iii) in Proposition \ref{klol322} of the Ito integral we can show the following relationship

Corollary. For any $X,Y\in {\cal L}_{t}^{2}$, we have
\[\mathbb{E}[I(X)\cdot I(Y)]=\int_{0}^{t}\mathbb{E}[X_{s}\cdot Y_{s}]ds. \sharp\]

So far we have only considered the Ito integral $I(X)$ of a function $X\in {\cal L}_{t}^{2}$ over a fixed time interval $[0,t]$. We shall continue to assume that $X\in {\cal L}_{t}^{2}$ and take any Borel subset $B$ of $[0,t]$. Then the Ito integral of $X$ over the subset $B$ is just the Ito integral $I(X\cdot 1_{B})$ of $X\cdot 1_{B}$ over $[0,t]$, where $1_{B}$ is the indicator function of $B$; clearly $X\cdot 1_{B}\in {\cal L}_{t}^{2}$. Usually we consider the subintervals $[t_{0},t_{1}]$ of $[0,t]$ and denote the resulting Ito integral by $\int_{t_{0}}^{t_{1}}XdW_{s}$. We could alternatively define this directly in terms of step functions defined only on $[t_{0},t_{1}]$.

For $0\leq t_{0}<t_{1}<t_{2}\leq t$, we have
\[X\cdot 1_{[t_{0},t_{2}]}=X\cdot 1_{[t_{0},t_{1}]}+X\cdot 1_{[t_{1},t_{2}]},\]
except at the instant $t_{1}$, and so from the linearity property (iv) in Proposition \ref{klol322}, we obtain
\begin{equation}{\label {kloeq3213}}\tag{19}
\int_{t_{0}}^{t_{2}}X_{s}(\omega )dW_{s}(\omega )=\int_{t_{0}}^{t_{1}}X_{s}(\omega )dW_{s}(\omega )+\int_{t_{1}}^{t_{2}}X_{s}\omega )dW_{s}(\omega ).\mbox{ a.s.}
\end{equation}

For a variable subinterval $[t_{0},s]\subseteq [0,t]$, we form a stochastic process $\{Z_{s}\}_{t_{0}\leq s\leq t}$, defined by
\begin{equation}{\label {kloeq3214}}\tag{20}
Z_{s}(\omega )=\int_{t_{0}}^{s}X_{u}(\omega )dW_{u}(\omega )\mbox{ a.s.}
\end{equation}
for $t_{0}\leq s\leq t$. Replacing $0$ by $t_{0}$ and $t$ by $s$ in Proposition \ref{klot323}, we have that $Z_{s}$ is ${\cal F}_{s}$-measurable with $\mathbb{E}[Z_{s}]=0$ and
\[\mathbb{E}[Z_{s}^{2}]=\int_{t_{0}}^{s}\mathbb{E}[X^{2}_{u}]du.\]
From (\ref{kloeq3213}) and (\ref{kloeq3214}) we then obtain for any $0\leq s’\leq s\leq t$
\[\mathbb{E}\left [\left |Z_{s}-Z_{s’}\right |^{2}\right ]=\int_{s’}^{s}\mathbb{E}[X^{2}_{u}]du,\]
from which it follows that $Z_{s}$ is mean-square continuous. Thus it has a separable and jointly ${\cal B}\times {\cal F}$-measurable version, which we shall use from now on. In fact, this version has, almost surely, continuous sample paths.

Proposition. For $t_{0}\leq u\leq s\leq t$, we have
\[\mathbb{E}[Z_{s}-Z_{u}|{\cal F}_{u}]=0\mbox{ a.s.}\]
i.e. $Z_{s}$ is a martingale. $\sharp$

Proposition. A separable, jointly measurable version of $Z_{s}$ defined by
\[Z_{s}(\omega )=\int_{t_{0}}^{s}X_{u}(\omega )dW_{u}(\omega )\]
for $s\in [t_{0},t]$ has, almost surely, continuous sample paths. $\sharp$

We recall that ${\cal F}(Z_{s})$ denotes the $\sigma$-field generated by the random variable $Z_{s}$. When $Z_{s}$ is defined by (\ref{kloeq3214}) it is ${\cal F}(Z_{s})$-measurable and ${\cal F}(Z_{s})\subseteq {\cal F}_{s}$. Thus from the properties of conditional expectations we have for any $t_{0}\leq t_{1}\leq t_{2}\leq t$
\[\mathbb{E}[Z_{t_{2}}-Z_{t_{1}}|{\cal F}(Z_{t_{1}})]=\mathbb{E}[\mathbb{E}[Z_{t_{2}}-Z_{t_{1}}|{\cal F}(Z_{t_{2}})]|{\cal F}(Z_{t_{1}})]=\mathbb{E}[0|{\cal F}(Z_{t_{1}})]=0\mbox{ a.s.}\]
since $Z_{s}$ is an ${\cal F}_{s}$-martingale. Consequently
\[\mathbb{E}[Z_{t_{2}}|{\cal F}(Z_{t_{1}})]=\mathbb{E}[Z_{t_{1}}|{\cal F}(Z_{t_{1}})]=Z_{t_{1}}\mbox{ a.s.}\]
so $Z_{s}$ is also an ${\cal F}(Z_{s})$-martingale.

Now we consider the case of change of time. In many ways a sample-path continuous version $Z_{s}$ of Ito integral (\ref{kloeq3214}) resembles a Brownian motion. It is ${\cal F}_{s}$-measurable with $Z_{t_{0}}=0$ and $\mathbb{E}[Z_{s}-Z_{u}|{\cal F}_{u}]=0$ a.s. for $t_{0}\leq u\leq s\leq t$. Then main difference is that we have
\[\mathbb{E}\left .\left [\left |Z_{s}-Z_{u}\right |^{2}\right |{\cal F}_{u}\right ]=\int_{0}^{s}\mathbb{E}\left .\left [X^{2}_{v}\right |{\cal F}_{u}\right ]dv\mbox{ a.s.}\]
instead of equalling $s-u$ as it would for a Brownian motion. When the integrand $X$ is nonrandom, that is $X_{t}(\omega )=x(t)$, we can transform the time variable to convert $Z$ into a Brownian time. To show this we shall suppose that $t_{0}=0$ and define
\[\tilde{s}(s)=\int_{0}^{s}x^{2}(u)du.\]
This is nondecreasing, but for simplicity we shall assume that it is strictly increasing and hence invertible. We define $\tilde{Z}_{\tilde{s}}=Z_{s(\tilde{s})}$ and $\tilde{{\cal F}}_{\tilde{s}}={\cal F}_{s(\tilde{s})}$, where $s(\tilde{s})$ is the inverse of $\tilde{s}(s)$. Then $\tilde{Z}_{\tilde{s}}$ is $\tilde{{\cal F}}_{\tilde{s}}$-measurable with $\tilde{Z}_{0}=0$, $\mathbb{E}[\tilde{Z}_{\tilde{s}}-\tilde{Z}_{\tilde{u}}| \tilde{{\cal F}}_{u}]=0$ and
\[\mathbb{E}\left .\left [\left |\tilde{Z}_{\tilde{s}}-\tilde{Z}_{\tilde{u}}\right |^{2}\right |\tilde{{\cal F}}_{u}\right ]=\int_{s}^{u}x^{2}(v)dv=\tilde{s}-\tilde{u}\mbox{ a.s.}\]
Hence the process $\{\tilde{Z}_{\tilde{s}}\}_{\tilde{s}\geq 0}$ is a Brownian motion with respect to the family of $\sigma$-fields $\{\tilde{{\cal F}}_{s}\}_{\tilde{s}\geq 0}$, at least for $0\leq\tilde{s}\leq\tilde{s}(t)$.

We shall extend the Ito integral to a wider class of integrands than those in the space ${\cal L}_{t}^{2}$. We shall say that $X$ belongs to ${\cal L}_{t}^{w}$ if $X$ is jointly ${\cal B}\times {\cal F}$-measurable, $X_{s}$ is ${\cal F}_{s}$-measurable for each $s\in [0,t]$ and
\[\int_{0}^{t}X^{2}_{s}(\omega )ds<\infty\mbox{ a.s.};\]
hence ${\cal L}_{t}^{2}\subset {\cal L}_{t}^{w}$. We then define $X^{(n)}\in {\cal L}_{t}^{2}$ by
\[X^{(n)}_{t}(\omega )=\left\{\begin{array}{ll}
X_{t}(\omega ) & \mbox{if }\int_{0}^{t}X^{2}_{s}(\omega )ds\leq n\\
0 & \mbox{otherwise}.
\end{array}\right .\]
The Ito stochastic integrals $I(X^{(n)})$ of the $X^{(n)}$ over $[0,t]$ are thus well-defined. It can then be shown that they converge in probability to a unique, a.s., random variable, which we shall denote by $I(X)$ and call the Ito stochastic integral of $X\in {\cal L}_{t}^{w}$ over the interval $[0.t]$. Apart from those properties explicitly involving expectations, which may now not exist, the Ito integrals of integrands $X\in {\cal L}_{t}^{w}$ satisfy analogous properties to those of integrands $X\in {\cal L}_{t}^{2}$. It is no longer a mean-square integrable martingale, but it is the convergence in probability limit of such martingales.

Ito’s Formula.

Let $\mu$ and $\sigma$ be two functions with $\sqrt{|\mu|}$ and $\sigma\in {\cal L}_{t}^{w}$, so that $\mu$ and $\sigma$ satisfy the properties required of a function in ${\cal L}_{t}^{w}$ except that the
integral of $\mu^{2}_{s}(\omega )$ is replaced by
\[\int_{0}^{t}|\mu_{s}(\omega )|ds<\infty\mbox{ a.s.}\]
Then, by a stochastic differential we mean an expression
\[dX_{s}(\omega )=\mu_{s}(\omega )ds+\sigma_{s}(\omega )dW_{s}(\omega ),\]
which is just a symbolical way of writing
\begin{equation}{\label {kloeq331}}\tag{21}
X_{t_{1}}(\omega )-X_{t_{0}}(\omega )=\int_{t_{0}}^{t_{1}}\mu_{s}(\omega )ds+\int_{t_{0}}^{t_{1}}\sigma_{s}(\omega )dW_{s}(\omega )\mbox{ a.s.}
\end{equation}
for any $0\leq t_{0}\leq t_{1}\leq t$.

Since the Ito integral for an integrand $\sigma\in {\cal L}_{t}^{w}$ is defined as the limit of Ito integrals for integrands in ${\cal L}_{t}^{2}$, we shall also consider the special case that $\sqrt{|\mu|}$ and
$\sigma\in {\cal L}_{t}^{2}$. In addition, we shall always suppose that $X_{s}$ is a separable, jointly measurable version of (\ref{kloeq331}) with, almost surely, continuous sample paths. When $\mu$ and $\sigma$ do not depend on $t$ they are ${\cal F}_{0}$-measurable random variables. The increments $X_{t_{1}}-X_{t_{0}}$ are then Gaussian and independent on non-overlapping intervals with $\mathbb{E}[X_{t_{1}}-X_{t_{0}}]=\mathbb{E}[\mu]\cdot (t_{1}-t_{0})$ and $\mbox{Var}[X_{t_{1}}-X_{t_{0}}]=\mathbb{E}[\sigma^{2}]\cdot (t_{1}-t_{0})$. In addition, from the linearity of the Lebesgue and Ito integrals, we have
\[d\left (\alpha X_{s}^{(1)}+\beta X_{s}^{(2)}\right )=\left (\alpha\cdot\mu_{s}^{(1)}+\beta\cdot\mu_{s}^{(2)}\right )ds+\left (\alpha\cdot\sigma_{s}^{(1)}+\beta\cdot\sigma_{s}^{(2)}\right )dW_{s}\] for any\ $\alpha ,\beta\in \mathbb{R}$, where
\[dX_{s}^{(i)}=\mu_{s}^{(i)}dt+\sigma_{s}^{(i)}dW_{s}\]
for $i=1,2$.

For nonlinear combinations or transformations of stochastic differentials we must use the Ito formula. The following result, which is a simple consequence of the Taylor and mean value Theorems of classical calculus, is useful for the derivation of Ito formula.

\begin{equation}{\label{klol331}}\tag{22}\mbox{}\end{equation}

Proposition \ref{klol331}. Let $U:[0,t]\times\mathbb{R}\rightarrow \mathbb{R}$ have continuous partial derivatives $\partial U/\partial s$, $\partial U/\partial x$ and $\partial^{2}U/\partial x^{2}$. Then for any $s,s+\Delta s\in [0,t]$ and $x,x+\Delta x\in \mathbb{R}$ there exist constants $0\leq\alpha\leq 1$ and $0\leq\beta\leq 1$ such that
\[U(s+\Delta s,x+\Delta x)-U(s,x)=\frac{\partial U}{\partial s}(s+\alpha\Delta s,x)\Delta s+\frac{\partial U}{\partial x}(s,x)\Delta x+
\frac{1}{2}\frac{\partial^{2}U}{\partial x^{2}}(s,x+\beta\Delta x)(\Delta x)^{2}. \sharp\]

In it we would gain little from the stronger assumption that $\sqrt{|\mu|},\sigma\in{\cal L}_{t}^{2}$ because that does not guarantee that the integrand $\sigma_{s}\cdot\frac{\partial U}{\partial x}(s,X_{s})$ of the Ito integral there belongs to ${\cal L}_{t}^{2}$ too; we can only conclude that it is in ${\cal L}_{t}^{w}$, but also holds when $\sqrt{|\mu|},\sigma\in {\cal L}_{t}^{w}$.

Theorem (Ito Formula). Let $Y_{s}=U(s,X_{s})$ for $0\leq s\leq t$ where $U$ is as in Proposition \ref{klol331} and $X_{s}$ satisfies (\ref{kloeq331}) with $\sqrt{|\mu|},\sigma\in {\cal L}_{t}^{w}$. Then
\[Y_{t_{1}}-Y_{t_{0}}=\int_{t_{0}}^{t_{1}}\left (\frac{\partial U}{\partial s}(s,X_{s})+\mu_{s}\cdot\frac{\partial U}{\partial x}(s,X_{s})+\frac{1}{2}
\sigma_{s}^{2}\cdot\frac{\partial^{2}U}{\partial x^{2}}(s,X_{s})\right )ds+\int_{t_{0}}^{t_{1}}\sigma_{s}\cdot\frac{\partial U}{\partial x}(s,X_{s})dW_{s}\mbox{ a.s.}\]
for any $0\leq t_{0}\leq t_{1}\leq t$. $\sharp$

Example. Let $dX_{s}=\sigma_{s}dW_{s}$ and $Y_{s}=U(s,X_{s})$. With $U(s,x)=e^{x}$, the Ito formula gives
\[dY_{s}=\frac{1}{2}\sigma_{s}^{2}Y_{s}ds+\sigma_{s}Y_{s}dW_{s},\]
whereas with
\[U(s,x)=\ex\mathbb{P}\left (x-\frac{1}{2}\int_{0}^{s}\sigma_{u}^{2}du\right )\]
it gives
\[dY_{s}=\sigma_{s}Y_{s}dW_{s}.\]
The later shows that the counterpart of the exponential in the Ito calculus is
\[\ex\mathbb{P}\left (X_{s}-\frac{1}{2}\int_{0}^{s}f_{u}^{2}du\right )=\ex\mathbb{P}\left (\int_{0}^{s}f_{u}dW_{u}-\frac{1}{2}\int_{0}^{s}f_{u}^{2}du\right )\]
rather than the $\exp (X_{s})$ of conventional calculus. $\sharP$

Now we consider the vector-valued Ito integral. Let $\{{\bf W}_{t}\}_{t\geq 0}$ be an $m$-dimensional Brownian motion with independent components associated with an increasing family of
$\sigma$-fields $\{{\cal F}_{t}\}_{t\geq 0}$. That is, ${\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(m)})$ where $W^{(j)}$ for $j=1,2,\cdots ,m$ are scalar Brownian motions with respect to
$\{{\cal F}_{t}\}_{t\geq 0}$, which are pairwise independent. Thus each $W_{t}^{(j)}$ is ${\cal F}_{t}$-measurable with $\mathbb{E}[W_{t}^{(j)}|{\cal F}_{0}]=0$ and $\mathbb{E}[W_{t}^{(j)}-W_{s}^{(j)}|{\cal F}_{s}]=0$ a.s. for $0\leq s\leq t$ and $j=1,2,\cdots ,m$. In addition,
\begin{equation}{\label {kloeq341}}\tag{23}
\mathbb{E}\left .\left [(W_{t}^{(i)}-W_{s}^{(i)})(W_{t}^{(j)}-W_{s}^{(j)})\right |{\cal F}_{s}\right ]=(t-s)\cdot\delta_{ij}\mbox{ a.s.}
\end{equation}
for $0\leq s\leq t$ and $i,j=1,2,\cdots ,m$.

We shall consider $d$-dimensional vector functions $\boldsymbol{\mu}:[0,t]\times\Omega\rightarrow \mathbb{R}^{d}$ with components $\mu^{(k)}$ satisfying $\sqrt{|\mu^{(k)}|}\in {\cal L}_{t}^{w}$ (or
${\cal L}_{t}^{2}$) for $k=1,2,\cdots ,d$ and $d\times m$-matrix functions $\boldsymbol{\sigma}:[0,t]\times\Omega\rightarrow\mathbb{R}^{d\times m}$ with components $\sigma^{(ij)}\in {\cal L}_{t}^{W}$ (or ${\cal L}_{t}^{2}$) for $k=1,2,\cdots ,d$ and $j=1,2,\cdots ,m$. Then we write symbolically as a $d$-dimensional vector stochastic differential
\begin{equation}{\label {kloeq342}}\tag{24}
d{\bf X}_{s}=\boldsymbol{\mu}_{s}ds+\boldsymbol{\sigma}_{s}d{\bf W}_{s}
\end{equation}
the vector stochastic integral expression
\[{\bf X}_{t_{1}}-{\bf X}_{t_{0}}=\int_{t_{0}}^{t_{1}}\boldsymbol{\mu}_{s}ds+\int_{t_{0}}^{t_{1}}\boldsymbol{\sigma}_{s}d{\bf W}_{s}\]
for any $0\leq t_{0}\leq t_{1}\leq t$, which we interpret componentwise as
\[X_{t_{1}}^{(k)}-X_{t_{0}}^{(k)}=\int_{t_{0}}^{t_{1}}\mu_{s}^{(k)}ds+\sum_{j=1}^{m}\int_{t_{0}}^{t_{1}}\sigma_{s}^{(kj)}dW_{s}^{(j)}\mbox{ a.s.}\]
for $k=1,2,\cdots ,d$. For a preassigned ${\cal F}_{0}$-measurable ${\bf X}_{0}$ the resulting $d$-dimensional stochastic process $\{{\bf X}_{t}\}_{t\geq 0}$ enjoys similar properties componentwise to
those listed in the previous discussions for scalar differentials involving a single Brownian motion, with additional properties relating the different components. The actual properties depend on whether the $\sqrt{|\mu^{(k)}|}$ and $\sigma^{(ij)}$ belong to ${\cal L}_{t}^{2}$ for all components or just to ${\cal L}_{t}^{w}$. In the former case with $\boldsymbol{\mu}={\bf 0}$, for example, we have $\mathbb{E}[X_{t_{1}}^{(k)}-X_{t_{0}}^{(k)}|{\cal F}_{t_{0}}]=0$ and
\begin{equation}{\label {kloeq345}}\tag{25}
\mathbb{E}\left .\left [(X_{t_{1}}^{(k)}-X_{t_{0}}^{(k)})(X_{t_{1}}^{(i)}-X_{t_{0}}^{(i)})\right |{\cal F}_{t_{0}}\right ]=\sum_{j=1}^{m}
\int_{t_{0}}^{t_{1}}\mathbb{E}[\sigma_{s}^{(kj)}\cdot\sigma_{s}^{(ij)}]ds\mbox{ a.s.}
\end{equation}
for $0\leq t_{0}\leq t_{1}\leq t$ and $k,i=1,2,\cdots ,d$. Here (\ref{kloeq345}) follows from the independence of the components of ${\bf W}$ and the identity (\ref{kloeq341}), which we could write symbolically
as $\mathbb{E}[dW_{s}^{(i)}dW_{s}^{(j)}]=\delta_{ij}ds$. As in the scalar case this leads to additional terms in the chain rule formula for the transformation of the vector stochastic differential (\ref{kloeq342}).

Let $U:[0,t]\times \mathbb{R}^{d}\rightarrow \mathbb{R}$ have continuous partial derivatives $\partial U/\partial s$, $\partial U/\partial x_{k}$ and $\partial^{2}U/\partial x_{i}\partial x_{j}$ for $i,j=1,2,\cdots ,d$, and define a scalar process $\{Y_{s}\}_{s\geq 0}$ by
\[Y_{s}=U(s,{\bf X}_{s})=U(s,X_{s}^{(1)},\cdots ,X_{s}^{(d)})\mbox{ a.s.}\]
where ${\bf X}_{s}$ satisfies the differential (\ref{kloeq342}). Then the stochastic differential for $Y_{s}$ is given by
\begin{equation}{\label {kloeq346}}\tag{26}
dY_{s}=\left (\frac{\partial U}{\partial s}+\sum_{k=1}^{d}\mu^{(k)}_{s}\cdot\frac{\partial U}{\partial x_{k}}+\frac{1}{2}\sum_{j=1}^{m}\sum_{k=1}^{d}
\sigma_{s}^{(ij)}\cdot\sigma_{s}^{(kj)}\cdot\frac{\partial^{2}U}{\partial x_{i}\partial x_{k}}\right )ds+\sum_{j=1}^{m}\sum_{i=1}^{d}\sigma_{s}^{(ij)}\cdot\frac{\partial U}{\partial x_{i}}dW_{s}^{(j)},
\end{equation}
where the partial derivatives are evaluated at $(s,{\bf X}_{s})$. This is the multicomponent analogue of the Ito formula, by which name it is also known. In vector-matrix notation it has the condensed form
\[dY_{s}=\left (\frac{\partial U}{\partial s}+\boldsymbol{\mu}^{T}_{s}\nabla U+\frac{1}{2}tr(\boldsymbol{\sigma}_{s}
\boldsymbol{\sigma}_{s}^{T}\nabla^{2}U)\right )ds+\nabla U^{T}\boldsymbol{\sigma}_{s}d{\bf W}_{s},\]
where $\nabla$ is the gradient operator and “$tr$” the trace of the matrix, that is the sum of its diagonal components. Thus $\nabla U$ is the vector of first order partial derivatives of $U$ and $\nabla^{2}U$ the matrix of the second partial order derivatives of $U$.

Example. Let $X_{s}^{(1)}$ and $X_{s}^{(2)}$ satisfy the scalar stochastic differentials
\begin{equation}{\label {kloeq348}}\tag{27}
dX_{s}^{(i)}=\mu_{s}^{(i)}dt+\sigma_{s}^{(i)}dW_{s}^{(i)}
\end{equation}
for $i=1,2$ and let $U(s,x_{1},x_{2})=x_{1}x_{2}$. Then the stochastic differential for the product process $Y_{s}=X_{s}^{(1)}\cdot X_{s}^{(2)}$ depends on whether the Brownian motions $W_{s}^{(1)}$ and $W_{s}^{(2)}$ are independent or dependent. In the former case the differentials (\ref{kloeq348}) can be written as the vector differential
\[d\left [\begin{array}{c} X_{s}^{(1)}\\ X_{s}^{(2)}\end{array}\right ]=\left [\begin{array}{c} \mu_{s}^{(1)}\\ \mu_{s}^{(2)}\end{array}\right ]+
\left [\begin{array}{cc} \sigma_{s}^{(1)} & 0\\ 0 & \sigma_{s}^{(2)}
\end{array}\right ]
d\left [\begin{array}{c} W_{s}^{(1)}\\ W_{s}^{(2)}\end{array}\right ]\]
and the transformed differential is
\[dY_{s}=(\mu_{s}^{(1)}X_{s}^{(2)}+\mu_{s}^{(2)}X_{s}^{(1)})ds+\sigma_{s}^{(1)}X_{s}^{(2)}dW_{s}^{(1)}+\sigma_{s}^{(2)}X_{s}^{(1)}dW_{s}^{(2)}.\]
In contrast, when $W_{s}^{(1)}=W_{s}^{(2)}=W_{s}$ the vector differential for (\ref{kloeq348}) is
\[d\left [\begin{array}{c} X_{s}^{(1)}\\ X_{s}^{(2)}\end{array}\right ]=\left [\begin{array}{c} \mu_{s}^{(1)}\\ \mu_{s}^{(2)}\end{array}\right ]+
\left [\begin{array}{c} \sigma_{s}^{(1)}\\ \sigma_{s}^{(2)}\end{array}\right ]dW_{s}\]
and there is an extra term $\sigma_{s}^{(1)}\sigma_{s}^{(2)}ds$ in the differential of $Y_{s}$, which is now
\begin{equation}{\label {kloeq3410}}\tag{28}
dY_{s}=(\mu_{s}^{(1)}X_{s}^{(2)}+\mu_{s}^{(2)}X_{s}^{(1)}+\sigma_{s}^{(1)}\sigma_{s}^{(2)})ds+(\sigma_{s}^{(1)}X_{s}^{(2)}+\sigma_{s}^{(2)}X_{s}^{(1)})dW_{s}. \sharp
\end{equation}

The Ito formula also holds for a vector-valued transformation ${\bf U}:[0,t]\times \mathbb{R}^{d}\rightarrow \mathbb{R}^{k}$ resulting in a vector-valued process ${\bf Y}_{s}={\bf U}(s,{\bf X}_{s})$. For
such processes the Ito formula (\ref{kloeq346}) is applied separately to each component $Y_{s}^{(i)}=U^{(i)}(s,{\bf X}_{s})$ for $i=1,2,\cdots ,k$.

Stratonovich Stochastic Integrals.

Now we are going to consider the other types of stochastic integrals. The Ito integral $\int_{0}^{t}X_{s}(\omega )dW_{s}(\omega )$ for an integrand $X\in {\cal L}_{t}^{2}$ is equal to the mean-square limit of the sums
\begin{equation}{\label {kloeq351}}\tag{29}
S_{n}(\omega )=\sum_{j=1}^{n}X_{\xi_{j}^{(n)}}(\omega )\cdot \left (W_{t_{j+1}^{(n)}}(\omega )-W_{t_{j}^{(n)}}(\omega )\right )
\end{equation}
with evaluation points $\xi_{j}^{(n)}=t_{j}^{(n)}$ for partitions $\pi =\left\{0=t_{1}^{(n)}<t_{2}^{(n)}<\cdots <t_{n+1}^{(n)}=t\right\}$ for which
\[\parallel\pi_{n}\parallel =\max_{1\leq j\leq n}\left (t_{j+1}^{(n)}-t_{j}^{(n)}\right )\rightarrow 0\mbox{ as }n\rightarrow\infty .\]
Other choices of evaluation points $t_{j}^{(n)}\leq\xi_{i}^{(n)}\leq t_{j+1}^{(n)}$ are possible, but generally lead to different random variables in the limit. While arbitrarily chosen evaluation points have little practical or theoretical use, those chosen systematically by
\begin{equation}{\label {kloeq352}}\tag{30}
\xi_{j}^{(n)}=(1-\lambda )t_{j}^{(n)}+\lambda t_{j+1}^{(n)}
\end{equation}
for the same fixed $0\leq\lambda\leq 1$ lead to limits, which we shall denote here by
\[(\lambda )\int_{0}^{t}X_{s}(\omega )dW_{s}(\omega ).\]
We note that the case $\lambda =0$ is just the Ito integral. The other case $0<\lambda\leq 1$ differ in that the process they define with respect to a variable upper integration endpoint is in general no loger a martingale.

When the integrand $X$ has continuously differentiable sample paths we obtain from Taylor’s Theorem
\[X_{(1-\lambda )t_{j}^{(n)}+\lambda t_{j+1}^{(n)}}(\omega )=(1-\lambda )\cdot X_{t_{j}^{(n)}}(\omega )+\lambda\cdot
X_{t_{j+1}^{(n)}}(\omega )+O\left (\left |t_{j+1}^{(n)}-t_{j}^{(n)}\right |\right ).\]
Since the higher order terms do not contribute to the limit as $\parallel\pi_{n}\parallel\rightarrow 0$, we see that $(\lambda )$-integral could then be evaluated alternatively as the mean-square limit of the sums
\begin{equation}{\label {kloeq353}}\tag{31}
S^{(\lambda)}_{n}(\omega )=\sum_{j=1}^{n}\left ((1-\lambda )\cdot X_{t_{j}^{(n)}}(\omega )+\lambda\cdot X_{t_{j+1}^{(n)}}(\omega )
\right )\cdot\left (W_{t_{j+1}^{(n)}}(\omega )-W_{t_{j}^{(n)}}(\omega )\right ).
\end{equation}
In the general case the $(\lambda )$-integrals are usually defined in terms of the sums (\ref{kloeq353}) rather than (\ref{kloeq351}) with evaluation points (\ref{kloeq352}), and we shall follow this practice here. As an indication of how the value of these integrals vary with $\lambda$ we observe that for $X_{s}(\omega )=W_{s}(\omega )$ we have
\begin{equation}{\label {kloeq354}}\tag{32}
(\lambda )\int_{0}^{t}W_{s}(\omega )dW_{s}(\omega )=\frac{1}{2}W_{t}^{2}(\omega )+\left (\lambda -\frac{1}{2}\right )t.
\end{equation}
This follows in the following mean-square limits
\[\sum_{j=1}^{n}W_{t_{j}^{(n)}}\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right )\rightarrow\frac{1}{2}W_{t}^{2}-\frac{1}{2}t\]
and
\begin{align*}
\sum_{j=1}^{n}W_{t_{j+1}^{(n)}}\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right ) & = & \sum_{j=1}^{n}\left (W_{t_{j+1}^{(n)}}-
W_{t_{j}^{(n)}}\right )^{2}+\sum_{j=1}^{n}W_{t_{j}^{(n)}}\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right )\\
& \rightarrow t+\left (\frac{1}{2}W_{t}^{2}-\frac{1}{2}t\right )=\frac{1}{2}W_{t}^{2}+\frac{1}{2}t,
\end{align*}
which are multiplied by $(1-\lambda )$ and $\lambda$, respectively, to give (\ref{kloeq354}). Unlike any of the others, the symmetric case $\lambda =1/2$ of the integral (\ref{kloeq354}), which was introduced by Stratonovich, does not contain term in addition to that given by classical calculus. It is now known as the Stratonovich integral and denoted by
\[\int_{0}^{t}X_{s}\circ dW_{s}\]
for any integrand $X\in {\cal L}_{t}^{2}$; it can be extended to integrands in ${\cal L}_{t}^{w}$ in the same way as for Ito integrals.

Usually only the Ito and Stratonovich integrals are widely used. As suggested in (\ref{kloeq354}) the Stratonovich integral obeys the transformation rules of classical calculus, and this is a major reason for its use. To see this, let $h:\mathbb{R}\rightarrow\mathbb{R}$ be continuously differentiable and consider the Stratonovich integral of $h(W_{t})$. By the Taylor formula we have
\[h\left (W_{t_{j+1}^{(n)}}\right )=h\left (W_{t_{j}^{(n)}}\right )+h’\left (W_{t_{j}^{(n)}}\right )\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right )+\mbox{higher order terms},\]
so the sum (\ref{kloeq353}) with $\lambda =1/2$ is
\begin{align*}
S^{(1/2)}_{n} & =\sum_{j=1}^{n}h\left (W_{t_{j}^{(n)}}\right )\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right )+\frac{1}{2}\sum_{j=1}^{n}h’\left (W_{t_{j}^{(n)}}\right )\cdot\left (W_{t_{j+1}^{(n)}}-W_{t_{j}^{(n)}}\right )^{2}+\mbox{higher order terms}\\
& \rightarrow & \int_{0}^{t}h(W_{s})dW_{s}+\frac{1}{2}\int_{0}^{t}h'(W_{s})ds
\end{align*}
in the mean-square sense. Hence
\begin{equation}{\label {kloeq355}}\tag{33}
\int_{0}^{t}h(W_{s})\circ dW_{s}=\int_{0}^{t}h(W_{s})dW_{s}+\frac{1}{2}\int_{0}^{t}h'(W_{s})ds.
\end{equation}
Now let $U$ be an anti-derivative of $h$, so $U'(x)=h(x)$ and hence $U”(x)=h'(x)$. Applying Ito’s formula to the transformation $Y_{s}=U(W_{s})$, we obtain
\[U(W_{t})-U(W_{0})=\frac{1}{2}\int_{0}^{t}h'(W_{s})ds+\int_{0}^{t}h(W_{s})dW_{s}.\]
Thus from (\ref{kloeq355}) the Stratonovich integral
\[\int_{0}^{t}h(W_{s})\circ dW_{s}=U(W_{t})-U(W_{0}),\]
as in classical calculus.

Example. For $h(x)=e^{x}$ an anti-derivative is $U(x)=e^{x}$, so
\[\int_{0}^{t}\exp (W_{s})\circ dW_{s}=\exp (W_{t})-1,\]
since $W_{0}=0$. Thus $Y_{s}=\exp (W_{s})$ is a solution of the Stratonovich stochastic differential equation $dY_{s}=Y_{s}\circ dW_{s}$. In contrast the Ito stochastic differential equation $dY_{s}=Y_{s}dW_{s}$ has the solution $Y_{s}=\exp (W_{s}-\frac{1}{2}t)$ for the same initial value $Y_{0}=1$. $\sharp$

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

Approach II (Convergence in L^{2}).

This approach follows from Oksendal \cite{oks}. Let $W$ be a Brownian motion. Suppose a stochastic process $X$ is given. We want to define the stochastic integral $\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )$. It is reasonable to start with a definition for a simple class of functions $X$ and then extend by some approximation procedure. Thus, let us first assume that $X$ has the form
\[X_{t}(\omega )=\sum_{j\geq 0}\xi_{j}(\omega )\cdot 1_{[t_{j},t_{j+1})}(t)\]
For such a function it is reasonable to define
\begin{equation}{\label {okseq318}}\tag{34}
\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )=\sum_{j\geq 0}\xi_{j}(\omega )\cdot (W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega )).
\end{equation}
However, without any further assumptions on the functions $\xi_{j}(\omega )$, this leads to difficulties, as the following example show.

Let us choose
\[X^{(1)}_{t}(\omega )=\sum_{j\geq 0}W_{t_{j}}(\omega )\cdot1_{[t_{j},t_{j+1})}(t)\mbox{ and }
X^{(2)}_{t}(\omega )=\sum_{j\geq 0}W_{t_{j+1}}(\omega )\cdot 1_{[t_{j},t_{j+1})}(t).\]
Then
\[\mathbb{E}\left [\int_{0}^{t}X^{(1)}_{u}(\omega )dW_{u}(\omega )\right ]=\sum_{j\geq 0}\mathbb{E}[W_{t_{j}}(W_{t_{j+1}}-W_{t_{j}})]=0,\]
since $\{W_{t}\}_{t\geq 0}$ has independent increment. But
\[\mathbb{E}\left [\int_{0}^{t}X^{(2)}_{u}(\omega )dW_{u}(\omega )\right ]=\sum_{j\geq 0}\mathbb{E}[W_{t_{j+1}}(W_{t_{j+1}}-W_{t_{j}})]=\sum_{j\geq 0}
\mathbb{E}[(W_{t_{j+1}}-W_{t_{j}})^{2}]=\sum_{j\geq 0}(t_{j+1}-t_{j})=t.\]
So, in spite of the fact that both $X^{(1)}$ and $X^{(2)}$ appear to be very reasonable approximations to $X_{t}(\omega )=W_{t}(\omega )$, their integrals according to (\ref{okseq318}) are not close each other at all no matter how large $n$ is chosen.

This only reflects the fact that the variations of the paths of $W_{t}$ are too big to enable us to define the stochastic integral in the Riemann-Steiltjes sense. In general, it is natural to approximate a given stocahstic process $X_{t}(\omega )$ by
\[\sum_{j}X_{t^{*}_{j}}(\omega )\cdot 1_{[t_{j},t_{j+1})}(t)\]
where the points $t_{j}^{*}$ belong to the intervals $[t_{j},t_{j+1}]$, and then define $\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )$ as the limit (in a sense that we will explain) of $\sum_{j}X_{t_{j}^{*}}(\omega ) (W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega ))$ as $n\rightarrow\infty$. However, the example above shows that, unlike the Riemann-Stieltjes integral, it does make a difference here what points $t_{j}^{*}$ we choose.

$t_{j}^{*}=t_{j}$, the left-hand point, which leads to the Ito integral, and denoted by
\[\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega );\]
$t_{j}^{*}=(t_{j}+t_{j+1})/2$, the mid point, which leads to the Stratonovich integral, and denoted by
\[\int_{s}^{t}X_{u}(\omega )\circ dW_{u}(\omega ).\]

We will here present Ito’s choice $t_{j}^{*}=t_{j}$. The approximation procedure indicated above will work out successfully provided that $X$ has the property that each of the functions $\omega\mapsto X_{t_{j}}(\omega )$ only depends on the behavior of $W_{s}(\omega )$ up to time $t_{j}$. This leads to the following important concepts.

Definition. Let ${\bf W}_{t}(\omega )$ be $n$-dimensional Brownian motion. Then we define ${\cal F}_{t}^{\bf W}$ to be the $\sigma$-filed generated by the random vectors ${\bf W}_{s}$ for $s\leq t$. In other words, ${\cal F}_{t}^{\bf W}$ is the smallest $\sigma$-filed containing all sets of the form $\{\omega :{\bf W}_{t_{1}}(\omega )\in F_{1},\cdots , {\bf W}_{t_{k}}(\omega ) \in F_{k}\}$, where $t_{j}\leq t$ and $F_{j}\subset \mathbb{R}^{n}$ are Borel sets, $j\leq k=1,2,\cdots$. We assume that all sets of measure zero are included in ${\cal F}_{t}^{\bf W}$. $\sharp$

One often thinks of ${\cal F}_{t}^{\bf W}$ as “the history of ${\bf W}_{s}$ up to time $t$”. We can show that a function $h(\omega )$ will be ${\cal F}_{t}^{\bf W}$-measurable if and only if $h$ can be written as the pointwise a.e. limit of sums of functions of the form
\[g_{1}({\bf W}_{t_{1}})\cdot g_{2}({\bf W}_{t_{2}})\cdots g_{k}({\bf W}_{t_{k}}),\]
where $g_{1},\cdots ,g_{k}$ are bounded continuous functions and $t_{j}\leq t$ for $j\leq k=1,2,\cdots$. Intuitively, that $h$ is ${\cal F}_{t}^{\bf W}$-measurable means that the value of $h(\omega )$ can be decided from the values of ${\bf W}_{s}(\omega )$ for $s\leq t$. For example, $h_{1}(\omega )={\bf W}_{t/2}(\omega )$ is ${\cal F}_{t}^{\bf W}$-measurable, while $h_{2}(\omega )={\bf W}_{2t}(\omega )$ is not (since $2t>t$). Thus the process $h_{1}(t,\omega )={\bf W}_{t/2}(\omega )$ is ${\cal F}_{t}^{\bf W}$-adapted, while $h_{2}(t,\omega )={\bf W}_{2t}(\omega )$ is not. We now describe the class of functions for which the Ito integral will be defined

\begin{equation}{\label{oksd314}}\tag{35}\mbox{}\end{equation}

Definition{\refoksd314}. Let ${\cal L}^{2}([s,t])$ be the class of stochastic processes $X_{u}(\omega )\equiv X(u,\omega ):[0,\infty)\times\Omega\rightarrow \mathbb{R}$ such that

$(u,\omega )\mapsto X_{u}(\omega )$ is ${\cal B}\times {\cal F}$, where ${\cal B}$ denotes the Borel $\sigma$-field on $[0,\infty )$.
$X_{u}(\omega )$ is ${\cal F}_{u}^{W}$-adapted.
${\displaystyle \mathbb{E}\left [\int_{s}^{t}X_{u}^{2}(\omega )du\right ]<\infty}$. $\sharp$

For stochastic processes $X\in {\cal L}^{2}([s,t])$ we will now show how to define the Ito integral
\[\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega ),\]
where $W_{u}$ is a Brownian motion. The idea is natural: First we define the integral for a simple class of stochastic processes $\phi$. Then we show that each $X\in {\cal L}^{2}([s,t])$ can be approximated (in an appropriate sense) by such $\phi$’s and we use this to define $\int_{s}^{t}XdW$ as the limit of $\int_{s}^{t}\phi dW$ as $\phi\rightarrow X$. A function $\phi\in {\cal L}^{2}([s,t])$ is called {\bf simple} if it has the form
\[\phi_{t}(\omega )=\sum_{j}\xi_{j}(\omega )\cdot 1_{[t_{j},t_{j+1})}(t).\]
Note that since $\phi\in {\cal L}^{2}([s,t])$ each function $\xi_{j}$ must be ${\cal F}_{t_{j}}^{W}$-measurable. For simple function $\phi_{t}(\omega )$ we define the integral according to (\ref{okseq318}), i.e.
\[\int_{s}^{t}\phi_{u}(\omega )dW_{u}(\omega )=\sum_{j\geq 0}\xi_{j}(\omega )(W_{t_{j+1}}(\omega )-W_{t_{j}}(\omega )).\]

\begin{equation}{\label{oksl315}}\tag{36}\mbox{}\end{equation}

Proposition \ref{oksl315}. (Ito Isometry). If $\phi_{t}(\omega )$ is a bounded and simple function, then
\begin{equation}{\label {okseq3111}}\tag{37}
\mathbb{E}\left [\left (\int_{s}^{t}\phi_{u}(\omega )dW_{u}(\omega )\right )^{2}\right ]=\mathbb{E}\left [\int_{s}^{t}\phi^{2}_{u}(\omega )du\right ].
\end{equation}

Proof. Put $\Delta W_{j}=W_{t_{j+1}}-W_{t_{j}}$. By independent increment, for $i<j$, $\Delta W_{i}$ is independent of $\Delta W_{j}$ and $\xi_{j}\in {\cal F}_{j}^{W}$ is independent of $\Delta W_{j}=W_{t_{j+1}}-W_{t_{j}}$ since the filtration ${\cal F}_{t}^{W}$ is generated by $\{W_{s}\}_{s\geq 0}$. Then
\[\mathbb{E}[\xi_{i}\xi_{j}\Delta W_{i}\Delta W_{j}]=\left\{\begin{array}{ll}
0 & \mbox{if $i\neq j$}\\
\mathbb{E}[\xi_{j}^{2}]\cdot (t_{j+1}-t_{j}) & \mbox{if $i=j$}.
\end{array}\right .\]
using the fact that $\xi_{i}\xi_{j}\Delta W_{i}$ and $\Delta W_{j}$ are independent if $i<j$. Thus
\[\mathbb{E}\left [\left (\int_{s}^{t}\phi dW\right )^{2}\right ]=\mathbb{E}\left [\left (\sum_{j}\xi_{j}\Delta W_{j}\right )^{2}\right ]=\sum_{i,j}\mathbb{E}\left [
\xi_{i}\xi_{j}\Delta W_{i}\Delta W_{j}\right ]=\sum_{j}\mathbb{E}[\xi_{j}^{2}]\cdot (t_{j+1}-t_{j})=\mathbb{E}\left [\int_{s}^{t}\phi^{2}_{u}du\right ].\]
This completes the proof. $\blacksquare$

The idea is now to use the isometry (\ref{okseq3111}) to extend the definition from simple stochastic processes to stochastic processes in ${\cal L}^{2}([s,t])$. We do this in several steps.

Step 1. Let $g\in {\cal L}^{2}([s,t])$ be bounded and $g_{u}(\omega )\equiv g(u,\omega )$ be continuous for each $\omega$. Then, there exist simple stochastic processes $latex \{\phi^{(n)}\}
\subset {\cal L}^{2}([s,t])$ satisfying
\[\mathbb{E}\left [\int_{s}^{t}\left (g_{u}-\phi^{(n)}_{u}\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow 0.\]
Proof. Define
\[\phi^{(n)}_{t}(\omega )=\sum_{j}g_{t_{j}^{(n)}}(\omega )\cdot 1_{[t_{j}^{(n)},t_{j+1}^{(n)})}(t).\]
Then $\phi^{(n)}$ is simple since $g\in {\cal L}^{2}([s,t])$. Since $g(t,\omega )$ is continuous for each $\omega$, we have
\[\int_{s}^{t}\left (g_{u}(\omega )-\phi^{(n)}_{u}(\omega )\right )^{2}du\rightarrow 0\mbox{ as }n\rightarrow\infty\mbox{ for each }\omega .\]
Hence
\[\mathbb{E}\left [\int_{s}^{t}\left (g_{u}-\phi^{(n)}_{u}\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow\infty\]
by bounded convergence.
Step 2. Let $h\in {\cal L}^{2}([s,t])$ be bounded. Then there exist bounded functions $g^{(n)}\in {\cal L}^{2}([s,t])$ such that $g^{(n)}_{u}(\omega )=g^{(n)}(u,\omega )$ is continuous for all $\omega$ and $n$, and
\[\mathbb{E}\left [\int_{s}^{t}\left (h_{u}-g^{(n)}_{u}\right )^{2}du\right ]\rightarrow 0.\]
Proof. Suppose $|h_{u}(\omega )|\leq M$ for all $(u,\omega )$. For each $n$ let $\psi_{n}$ be a nonnegative and continuous function on $\mathbb{R}$ satisfying
\[\psi_{n}(x)=0\mbox{ for }x\leq -\frac{1}{n}\mbox{ and }x\geq 0,\mbox{ and }\int_{-\infty}^{\infty}\psi_{n}(x)dx=1.\]
Define
\[g^{(n)}_{u}(\omega )=g^{(n)}(u,\omega )=\int_{0}^{u}\psi_{n}(v-u)h_{v}(\omega )dv.\]
Then $g^{(n)}(u,\omega )$ is continuous for each $\omega$ and
\[|g^{(n)}_{u}(\omega )|\leq\int_{0}^{u}|\psi_{n}(v)||h_{v}(\omega )|dv\leq M\int_{0}^{u}|\psi_{n}(v)|dv=M\int_{0}^{u}\psi_{n}(v)dv\leq M\int_{-\infty}^{\infty}\psi_{n}(v)dv=M.\]
Since $h\in {\cal L}^{2}([s,t])$ we see that $g^{(n)}_{u}$ is ${\cal F}_{u}$-measurable for all $u$. (Use sums to approximate the integral defining $g^{(n)}$). Moreover,
\[\int_{s}^{t}\left (g^{(n)}_{u}(\omega )-h_{u}(\omega )\right )^{2}du\rightarrow 0\mbox{ as }n\rightarrow\infty\mbox{ for each }\omega\]
since $\{\psi_{n}\}$ constitutes an approximate identity (See Hoffmann,\cit\mathbb{E}[p.22]{hof62}). So by bounded convergence
\[\mathbb{E}\left [\int_{s}^{t}\left (h_{u}(\omega )-g^{(n)}_{u}(\omega )\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow\infty\]
as asserted.
Step 3. Let $X\in {\cal L}^{2}([s,t])$. Then there exists a sequence $\{h^{(n)}\}\subset {\cal L}^{2}([s,t])$ such that $h^{(n)}$ is bounded for each $n$ and
\[\mathbb{E}\left [\int_{s}^{t}\left (X_{u}-h^{(n)}_{u}\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow\infty .\]
Proof.
Put
\[h^{(n)}_{u}(\omega )=\left\{\begin{array}{ll}
-n & \mbox{if $X_{u}(\omega )<-n$}\\
X_{u}(\omega ) & \mbox{if $-n\leq X_{u}(\omega )\leq n$}\\
n & \mbox{if $X_{u}(\omega )>n$}.
\end{array}\right .\]
Then the conclusion follows by dominated convergence.

That completes the approximation procedure. We are now ready to complete the definition of the Ito integral
\[\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )\mbox{ for }X\in{\cal L}^{2}([s,t]).\]
If $X\in {\cal L}^{2}([s,t])$ we choose, by Steps 1-3, simple stochastic processes $\{\phi^{(n)}\}\subset {\cal L}^{2}([s,t])$ such that
\[\mathbb{E}\left [\int_{s}^{t}\left (X_{u}-\phi^{(n)}_{u}\right )^{2}dt\right ]\rightarrow 0.\]
Now we let
\[y_{n}(\omega )=\int_{s}^{t}\phi^{(n)}_{u}(\omega )dW_{u}(\omega )\]
Then, we have
\begin{align*}
\mathbb{E}[(y_{n}-y_{m})^{2}] & =\mathbb{E}\left [\left (\int_{s}^{t}\phi^{(n)}_{u}dW_{u}-\int_{s}^{t}\phi^{(m)}_{u}dW_{u}\right )^{2}\right ]=\mathbb{E}\left [\left (
\int_{s}^{t}(\phi^{(n)}_{u}-\phi^{(m)}_{u})dW_{u}\right )^{2}\right ]\\
& = &\mathbb{E}\left [\int_{s}^{t}\left (\phi^{(n)}_{u}-\phi^{(m)}_{u}\right )^{2}du\right ]
\end{align*}
by (\ref{okseq3111}). Furthermore we see that
\[\mathbb{E}\left [\int_{s}^{t}\left (\phi^{(n)}_{u}-\phi^{(m)}_{u}\right )^{2}du\right ]\rightarrow 0;\]
that is, $\{y_{n}\}=\left\{\int_{s}^{t}\phi^{(n)}_{u}dW_{u}\right\}$ forms a Cauchy sequence in $L^{2}(P)$. Since $L^{2}(P)$ is complete space, we define
\[\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )=\lim_{n\rightarrow\infty}\int_{s}^{t}\phi^{(n)}_{u}(\omega )dW_{u}(\omega )\]
as the element in $L^{2}(P)$.

Definition. Let $X\in {\cal L}^{2}([s,t])$. Then the Ito integral of $X$ is defined by
\begin{equation}{\label {okseq3112}}\tag{38}
\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )=\lim_{n\rightarrow\infty}\int_{s}^{t}\phi^{(n)}_{u}(\omega )dW_{u}(\omega )
\end{equation}
The limit in $L^{2}(P)$, where $\{\phi^{(n)}\}$ is a sequence of simple stochastic processes such that
\begin{equation}{\label {okseq3113}}\tag{39}
\mathbb{E}\left [\int_{s}^{t}\left (X_{u}(\omega )-\phi^{(n)}_{u}(\omega )\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow\infty . \sharp
\end{equation}

Note that such a sequence $\{\phi^{(n)}\}$ satisfying (\ref{okseq3113}) exists by Steps 1-3 above. Moreover, by (\ref{okseq3111}), the limit in (\ref{okseq3112}) exists and does not depend on the actual choice of
$\{\phi^{(n)}\}$ as long as (\ref{okseq3113}) holds. Furthermore, from (\ref{okseq3111}) and (\ref{okseq3112}) we get the following result.

Proposition. (Ito Isometry). We have
\[\mathbb{E}\left [\left (\int_{s}^{t}X_{u}(\omega )dW_{u}\right )^{2}\right ]=\mathbb{E}\left [\int_{s}^{t}X^{2}_{u}(\omega )du\right ]\]
for all $X\in {\cal L}^{2}([s,t])$. $\sharp$

\begin{equation}{\label{oksc318}}\tag{40}\mbox{}\end{equation}

Proposition.\refksc318}. If $X\in {\cal L}^{2}([s,t])$ and $X^{(n)}\in {\cal L}^{2}([s,t])$ for $n=1,2,\cdots$ and
\[\mathbb{E}\left [\int_{s}^{t}\left (X^{(n)}_{u}(\omega )-X_{u}(\omega )\right )^{2}du\right ]\rightarrow 0\mbox{ as }n\rightarrow\infty ,\]
then
\[\int_{s}^{t}X^{(n)}_{u}(\omega )dW_{u}(\omega )\rightarrow\int_{s}^{t}X_{u}(\omega )dW_{u}(\omega )\]
in $L^{2}(P)$ as $n\rightarrow\infty$. $\sharp$

\begin{equation}{\label{okst321}}\tag{41}\mbox{}\end{equation}

Proposition \ref{okst321}. Let $X,Y\in {\cal L}^{2}([0,t])$ and let $0\leq s<v<t$. Then, we have the following properties.

(i) ${\displaystyle \int_{s}^{t}X_{u}dW_{u}=\int_{s}^{v}X_{u}dW_{u}+\int_{v}^{t}X_{u}dW_{u}}$ a.s.

(ii) ${\displaystyle \int_{s}^{t}(cX_{u}+Y_{u})dW_{u}=c\int_{s}^{t}X_{u}dW_{u}+\int_{s}^{t}Y_{u}dW_{u}}$ a.s.

(iii) ${\displaystyle \mathbb{E}\left [\int_{s}^{t}X_{u}dW_{u}\right ]=0}$.

(iv) ${\displaystyle \int_{s}^{t}X_{u}dW_{u}}$ is ${\cal F}_{t}$-measurable. $\sharp$

Theorem. (Doob’s Martingale Inequality). If $M_{u}$ is a martingale such that $t\mapsto M_{u}(\omega )$ is continuous a.s., then for all $p\geq 1$, $t\geq 0$ and all $\lambda >0$
\[\mathbb{P}\left\{\sup_{0\leq u\leq t}|M_{u}|\geq\lambda\right\}\leq\frac{1}{\lambda^{p}}\cdot \mathbb{E}[|M_{u}|^{p}]. \sharp\]

We can use this inequality to prove that the Ito integral $\int_{0}^{s}X_{u}(\omega )dW_{u}$ can be chosen to depend continuously on $s$.

Proposition. Let $X\in {\cal L}^{2}([0,t])$. Then there exists a $s$-continuous version of $\int_{0}^{s}X_{u}(\omega )dW_{u}(\omega )$ for $0\leq s\leq t$, i.e. there exists a $s$-continuous stochastic process $J_{s}$ on $(\Omega ,{\cal F},P)$ satisfying
\[\mathbb{P}\left\{J_{s}=\int_{0}^{s}XdW\right\}=1\mbox{ for all }0\leq s\leq t. \sharp\]

From now on we shall always assume that $\int_{0}^{s}X_{u}(\omega )dW_{u}(\omega )$ means a $s$-continuous version of the integral.

\begin{equation}{\label{oksc*1}}\tag{42}\mbox{}\end{equation}

Corollary \ref{oksc*1}. Let $X\in {\cal L}^{2}([0,t])$ for all $t$. Then $M_{s}=\int_{0}^{s}X_{u}(\omega )dW_{u}$ is a martingale wit respect to ${\cal F}_{s}^{W}$ and
\[\mathbb{P}\left\{\sum_{0\leq s\leq t}|M_{s}|\geq\lambda\right\}\leq\frac{1}{\lambda^{2}}\cdot \mathbb{E}\left [\int_{0}^{s}X^{2}_{u}(\omega )du\right ]\mbox{ for }\lambda ,t>0. \sharp\]

The Ito integral can be defined for a larger class of integrands $X$ than\ ${\cal L}^{2}([s,t])$. First, the measurability condition (ii) of Definition~\ref{oksd314} can be relaxed to the following

$\mbox{(ii)}’$ There exists an increasing family of $\sigma$-fileds $\{{\cal H}_{t}\}_{t\geq 0}$ such that

$W_{t}$ is a martingale with respect to ${\cal H}_{t}$ and
$X_{t}$ is ${\cal H}_{t}$-adapted.

Note that (a) implies that ${\cal F}_{t}^{W}\subset {\cal H}_{t}$ (${\cal F}_{t}^{W}$ is generated by $\{W_{t}\}_{t\geq 0}$). The essence of this extension is that we can allow $X_{t}$ to depend on more than ${\cal F}_{t}^{W}$ as long as $W_{t}$ remains a martingale with respect to the “history” of $X_{s}$ for $s\leq t$. If $\mbox{(ii)}’$ holds, then $\mathbb{E}[W_{s}-W_{t}|{\cal H}_{t}]=0$ for all $s>t$ and if we inspect the proofs above, we see that this is sufficient to carry out the construction of the Ito integral as before.

Suppose $W^{(k)}_{t}(\omega )$ is the $k$th coordinate of $n$-dimensional Brownian motion $(W^{(1)},\cdots ,W^{(n)})$. Let ${\cal F}_{t}^{\bf W}$ be the $\sigma$-field generated by $W^{(1)}_{s_{1}}(\omega ),\cdots ,W^{(n)}_{s_{n}}(\omega )$ for $s_{k}\leq t$. Then $W^{(k)}_{t}(\omega )$ is a martingale with respect to ${\cal F}_{t}^{\bf W}$ because $W^{(k)}_{s}(\omega )-W^{(k)}_{t}(\omega )$ is independent of ${\cal F}_{t}^{\bf W}$ when $s>t$. Thus we have now defined $\int_{s}^{t}X_{u}(\omega )dW^{(k)}_{u}(\omega )$ for ${\cal F}_{t}^{\bf W}$-adapted integrands $X_{t}(\omega )$. That includes integrals like
\[\int W^{(2)}dW^{(1)}\mbox{ or }\int\sin \left ((W^{(1)})^{2}+(W^{(2)})^{2}\right )dW^{(2)}\]
involving several components of $n$-dimensional Brownian motion. This allows us to define the {\bf multi-dimensional Ito Integral} as follows.

Definition. Let ${\bf W}=(W^{(1)},W^{(2)},\cdots ,W^{(n)})$ be $n$-dimensional Brownian motion. Then ${\cal L}_{{\cal H}}^{m\times n}([s,t])$ denotes the set of $m\times n$ matrices ${\bf v}=[v^{(ij)}_{t}(\omega )]$ where each entry $v^{(ij)}_{t}(\omega )$ satisfies (i) and (iii) of Definition \ref{oksd314} and (ii)’ above with respect to some filtration $\{{\cal H}_{t}\}_{t\geq 0}$. If $latex {\bf v}\in
{\cal L}_{{\cal H}}^{m\times n}([s,t])$ we define, using matrix notation
\[\int_{s}^{t}{\bf v}d{\bf W}=\int_{s}^{t}
\left [\begin{array}{ccc}
v^{(11)} & \cdots & v^{(1n)}\\
\vdots & \vdots & \vdots\\
v^{(m1)} & \cdots & v^{(mn)}
\end{array}\right ]
\left [\begin{array}{c}
dW^{(1)}\\ \vdots \\ dW^{(n)}
\end{array}\right ]\]
to be the $m\times 1$ matrix whose $i$th component is the following sum of $1$-dimensional Ito integrals
\[\sum_{j=1}^{n}\int_{s}^{t}v^{(ij)}_{u}(\omega )dW^{(j)}_{u}(\omega ).\]
If ${\cal H}=\{{\cal F}_{t}^{\bf W}\}_{t\geq 0}$ we write ${\cal L}^{m\times n}([s,t])$ and if $m=1$ we write ${\cal L}_{{\cal H}}^{n}([s,t])$ $($resp. ${\cal L}^{n}([s,t]))$ instead of ${\cal L}_{{\cal H}}^{n\times 1}([s,t])$ $($resp. ${\cal L}^{n\times 1}([s,t]))$. We also put
\[{\cal L}^{m\times n}={\cal L}^{m\times n}(0,\infty )=\bigcap_{t>0}{\cal L}^{m\times n}([0,t]). \sharp\]

The next extension of the Ito integral consists of weakening condition (iii) of Definition \ref{oksd314} to

(iii)’ ${\displaystyle \mathbb{P}\left\{\int_{s}^{t}X^{2}_{u}(\omega )du<\infty\right\}=1.}$

\begin{equation}{\label{oksd*1}}\tag{43}\mbox{}\end{equation}

Definition \ref{oksd*1}. ${\cal W}_{{\cal H}}([s,t])$ denotes the class of processes $X_{u}(\omega )\in \mathbb{R}$ satisfying (i) of Definition \ref{oksd314} and (ii)’ and (iii)’$ above. Similarly to the notation for ${\cal L}$ we put ${\cal W}_{{\cal H}}=\bigcup_{t>0} {\cal W}_{{\cal H}}([0,t])$ and in the matrix case we write ${\cal W}_{{\cal H}}^{m\times n}([s,t])$ etc. If ${\cal H}=\{{\cal F}_{t}^{\bf W}\}_{t\geq 0}$ we write ${\cal W}([s,t])$ instead of ${\cal W}_{{\cal F}^{W}}([s,t])$ etc. $\sharp$

Let $W$ be a $1$-dimensional Brownian motion. If $X\in {\cal W}_{{\cal H}}$ one can show that for all $t$ there exist step functions $X^{(n)}\in {\cal W}_{{\cal H}}$ such that
\[\int_{0}^{t}\left |X^{(n)}_{s}(\omega )-X_{s}(\omega )\right |^{2}ds\rightarrow 0\mbox{ {\bf in probability}}.\]
For such sequence one has that $\int_{0}^{t}X^{(n)}_{s}(\omega )dW_{s}(\omega )$ converges in probability to some random variable and the limit only depends on $X$, not on the sequence $\{X^{(n)}\}$. Thus we may define
\[\int_{0}^{t}X_{s}(\omega )dW_{s}(\omega )=\lim_{n\rightarrow\infty}\int_{0}^{t}X^{(n)}_{s}(\omega )dW_{s}(\omega )\]
limit in probability for $X\in {\cal W}_{{\cal H}}$. As before there exists a $t$-continuous version of this integral. Note, however, that this integral is not in general a martingale. It is, however, a local martingale.

Ito’s Formula.

\begin{equation}{\label{oksd411}}\tag{44}\mbox{}\end{equation}

Definition \ref{oksd411}. Let $W$ be one-dimensional Brownian motion on $(\Omega ,{\cal F},\mathbb{P})$. A one-dimensional Ito process is a stochastic process $X$ on $(\Omega ,{\cal F},\mathbb{P})$ of the form
\begin{equation}{\label {okseq413}}\tag{45}
X_{t}=X_{0}+\int_{0}^{t}\mu_{s}(\omega )ds+\int_{0}^{t}\sigma_{s}(\omega )dW_{s},
\end{equation}
where $\sigma\in {\cal W}_{{\cal H}}$ $($see Definition~\ref{oksd*1}$)$, so that
\[\mathbb{P}\left\{\int_{0}^{t}\sigma_{s}^{2}(\omega )ds<\infty\mbox{ for all }t\geq 0\right\}=1\]
We also assume that $\mu$ is ${\cal H}_{t}$-adapted, where ${\cal H}_{t}$ is as in $\mbox{{\em (ii)}}’$ below Corollary~\ref{oksc*1}, and
\[\mathbb{P}\left\{\int_{0}^{t}|\mu_{s}(\omega )|ds<\infty\mbox{ for all }t\geq 0\right\}=1. \sharp\]

If $X$ is an Ito process of the form (\ref{okseq413}), the equation (\ref{okseq413}) is sometimes written in the shorter differential form
\[dX_{t}=\mu_{t}dt+\sigma_{t}dW_{t}.\]

Theorem. (The one-dimensinal Ito Formula). Let $X$ be an Ito process given by $dX_{t}=\mu_{t}dt+\sigma_{t}dW_{t}$. Let $g(t,x)\in C^{2}([0,\infty )\times \mathbb{R})$, i.e., $g$ is twice continuously differentiable on $[0,\infty )\times \mathbb{R}$. Then $Y_{t}=g(t,X_{t})$ is again an Ito process, and
\[dY_{t}=\frac{\partial g}{\partial t}(t,X_{t})dt+\frac{\partial g}{\partial x}(t,X_{t})dX_{t}+\frac{1}{2}\cdot\frac{\partial^{2}g}{\partial x^{2}}(t,X_{t})(dX_{t})^{2},\]
where $(dX_{t})^{2}=(dX_{t})(dX_{t})$ is computed according to the rules $dt\cdot dt=dt\cdot dW_{t}=dW_{t}\cdot dt=0$ and $dW_{t}\cdot dW_{t}=t$. $\sharp$

Example. Let us consider the integral
\[\int_{0}^{t}W_{s}dW_{s}.\]
Choose $X_{t}=W_{t}$ and $g(t,x)=x^{2}/2$. Then $Y_{t}=g(t,W_{t})=(W_{t})^{2}/2$. Then by Ito’s formula
\[dY_{t}=\frac{\partial g}{\partial t}dt+\frac{\partial g}{\partial x}dW_{t}+\frac{1}{2}\cdot\frac{\partial^{2}g}{\partial x^{2}}(dW_{t})^{2}=W_{t}dW_{t}+\frac{1}{2}t.\]
hence
\[d\left (\frac{1}{2}W_{t}^{2}\right )=W_{t}dW_{t}+\frac{1}{2}t.\]
In other words,
\[\frac{1}{2}W_{t}^{2}=\int_{0}^{t}W_{s}dW_{s}+\frac{1}{2}t. \sharp\]

Example. Let us consider the intergal
\[\int_{0}^{t}sdW_{s}.\]
From classical calculus it seems reasonable that a term of the form $tW_{t}$ should appear, so we put $g(t,x)=tx$ and $Y_{t}=g(t,W_{t})=tW_{t}$. Then by Ito’s formula
\[dY_{t}=W_{t}dt+tdW_{t}+0,\mbox{ that is, }d(tW_{t})=W_{t}dt+tdW_{t}\]
or
\[tW_{t}=\int_{0}^{t}W_{s}ds+\int_{0}^{t}sdW_{s}\]
or
\[\int_{0}^{t}sdW_{s}=tW_{t}-\int_{0}^{t}W_{s}ds,\]
which is reasonable from an integration-by-parts point of view. $\sharp$

\begin{equation}{\label{okst415}}\tag{46}\mbox{}\end{equation}

Theorem \ref{okst415}. (Integration by Parts). Suppose $X_{s}(\omega )=f(s)$ only depends on $s$ and that $f$ is continuous and of bounded variation on $[0,t]$. Then
\[\int_{0}^{t}X_{s}(\omega )dW_{s}(\omega )=\int_{0}^{t}f(s)dW_{s}(\omega )=f(t)W_{t}(\omega )-\int_{0}^{t}W_{s}(\omega )df_{s}. \sharp\]

Note that it is crucial for the result to hold that $X_{s}(\omega )=f(s)$ does not depend on $\omega$.

We now turn to the situation in higher dimensions. Let ${\bf W}_{t}(\omega )=(W^{(1)}_{t}(\omega ),\cdots ,W^{(m)}_{t}(\omega ))$ denotes $m$-dimensional Brownian motion. If each of the processes $\mu^{(i)}_{t}(\omega )$ and $\sigma^{(ij)}_{t}(\omega )$ satisfies the conditions given in Definition \ref{oksd411}, $1\leq i\leq n$, $1\leq j\leq m$, then we can form the following $n$ Ito processes
\[\begin{array}{ccc}
dX^{(1)} & = & \mu^{(1)}dt+\sigma^{(11)}dW^{(1)}+\cdots
+\sigma^{(1m)}dW^{(m)}\\
\vdots && \vdots\\
dX^{(n)} & = & \mu^{(n)}dt+\sigma^{(n1)}dW^{(1)}+\cdots +\sigma^{(nm)}dW^{(m)}
\end{array}\]
Or in matrix notation simply
\[d{\bf X}_{t}=\boldsymbol{\mu}_{t}dt+\boldsymbol{\sigma}_{t}d{\bf W}_{t},\]
where
\[{\bf X}_{t}=\left [\begin{array}{c}X^{(1)}_{t}\\ \vdots \\ X^{(n)}_{t}
\end{array}\right ],
\boldsymbol{\mu}_{t}=\left [\begin{array}{c}
\mu^{(1)}_{t}\\ \vdots \\ \mu^{(n)}_{t}
\end{array}\right ],
\boldsymbol{\sigma}_{t}=\left [\begin{array}{ccc}
\sigma^{(11)}_{t} & \cdots & \sigma^{(1m)}_{t}\\
\vdots & \vdots & \vdots\\
\sigma^{(n1)}_{t} & \cdots & \sigma^{(nm)}_{t}
\end{array}\right ],
d{\bf W}_{t}=\left [\begin{array}{c}
dW^{(1)}_{t}\\ \vdots \\dW^{(m)}_{t}
\end{array}\right ].\]
Such a process ${\bf X}_{t}$ is called an $n$-dimensional Ito process.

\begin{equation}{\label{okst421}}\tag{47}\mbox{}\end{equation}

Theorem \ref{okst421}. (Ito Formula) Let $d{\bf X}_{t}=\boldsymbol{\mu}_{t}dt+\boldsymbol{\sigma}_{t}d{\bf W}_{t}$ be an $n$-dimensional Ito process as above. Let ${\bf g}(t,x)=(g_{1}(t,x),\cdots ,g_{d}(t,x))$ be a $C^{2}$ map from $[0,\infty )\times \mathbb{R}^{n}$ into $\mathbb{R}^{d}$. Then the process ${\bf Y}_{t}={\bf g}(t,{\bf X}_{t})$ is again an Ito process, whose $k$th component $Y^{(k)}$ is given by
\[dY^{(k)}_{t}=\frac{\partial g_{k}}{\partial t}(t,{\bf X}_{t})dt+\sum_{i}\frac{\partial g_{k}}{\partial x_{i}}(t,{\bf X}_{t})dX^{(i)}+
\frac{1}{2}\sum_{i,j}\frac{\partial^{2}g_{k}}{\partial x_{i}\partial x_{j}}(t,{\bf X}_{t})dX^{(i)}dX^{(j)}\]
where $dW^{(i)}dW^{(j)}=\delta_{ij}dt$ and $dW^{(i)}dt=dtdW^{(i)}=0$. $\sharp$

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

Approach III (Convergence in Probability)

This follows from Klebaner \cite{kle}. Consider first integrals with deterministic simple process $X_{s}$, which is a function of $t$ and does not depend on $W_{s}$. By definition a simple deterministic process $X_{s}$ is such a process for which there exist times $0=t_{0}<t_{1}<\cdots <t_{n}=t$ and constants $c_{0},c_{1},\cdots , c_{n-1}$ satisfying
\[X_{s}=\left\{\begin{array}{ll}
c_{0} & \mbox{if $s=0$}\\
c_{i} & \mbox{if $t_{i}<s\leq t_{i+1}, i=0,1,\cdots ,n-1$,}
\end{array}\right .\]
or in one formula
\[X_{s}=c_{0}\cdot 1_{\{0\}}(s)+\sum_{i=1}^{n-1}c_{i}\cdot 1_{(t_{i},t_{i+1}]}(s).\]
The Ito integral $\int_{0}^{t}X_{s}dW_{s}$ is defined as a sum
\begin{equation}{\label {kleeq42}}\tag{48}
\int_{0}^{t} X_{s}dW_{s}=\sum_{i=1}^{n-1}c_{i}\cdot (W_{t_{i+1}}-W_{t_{i}}).
\end{equation}
It is easy to see by using the independence property of Brownian increments that the integral, which is the sum in (\ref{kleeq42}) is a Gaussian random variable with mean zero and variance
\begin{align*}
Var\left (\int_{0}^{t}X_{s}dW_{s}\right ) & =\mathbb{E}\left [\left (\sum_{i=0}^{n-1}c_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\right )^{2}\right ]\\
& =\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\mathbb{E}\left [c_{i}c_{j}\cdot (W_{t_{i+1}}-W_{t_{i}})\cdot (W_{t_{j+1}}-W_{t_{j}})\right ]=\sum_{i=1}^{n-1}c_{i}^{2}\cdot (t_{i+1}-t_{i}).
\end{align*}
Since we would like to integrate stochastic processes, it is important to allow for constants $c_{i}$ to be random. If $c_{i}$’s are replaced by random variables $\xi_{i}$’s, then in order to carry out calculations, and have convenient properties of the integral, the random variables $\xi_{i}$’s are allowed to depend on the values of $W_{s}$ for $s<t_{i}$, that is , they are allowed to be ${\cal F}_{t_{i}}^{W}$-measurable, where ${\cal F}_{t_{i}}^{W}$ is the $\sigma$-field generated by Brownian motion up to time $t_{i}$. Indeed, if $\xi_{i}$’s are ${\cal F}_{t_{i}}$-measurable, then by the martingale property of Brownian motion
\[\mathbb{E}\left .\left [\xi_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\right |{\cal F}_{t_{i}}\right ]=\xi_{i}\cdot \mathbb{E}\left .\left [(W_{t_{i+1}}-W_{t_{i}})\right |{\cal F}_{t_{i}}\right ]=0,\]
and it follows that
\begin{equation}{\label {kleeq44}}\tag{49}
\mathbb{E}\left [\xi_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\right ]=0.
\end{equation}

Let $X=\{X_{s}\}_{0\leq s\leq t}$ be a process for which there exist times $0=t_{0}<t_{1}<\cdots <t_{n}=t$ and random variables $\xi_{0},\xi_{1},\cdots ,\xi_{n-1}$ such that $\xi_{0}$ is a constant, $\xi_{i}$ depends on the values of $W_{s}$ for $s\leq t_{i}$, but not on values of $W_{s}$ for $s>t_{i}$, $i=0,\cdots ,n-1$; and
\[X_{s}=\xi_{0}\cdot 1_{\{0\}}(s)+\sum_{i=1}^{n-1}\xi_{i}\cdot 1_{(t_{i},t_{i+1}]}(s)\]
such processes are a particular case of simple predictable processes. Ito integral $\int_{0}^{t}X_{s}dW_{s}$ is defined as a sum
\begin{equation}{\label {kleeq46}}\tag{50}
\int_{0}^{t}X_{s}dW_{s}=\sum_{i=1}^{n-1}\xi_{i}\cdot (W_{t_{i+1}}-W_{t_{i}}).
\end{equation}
The Ito integral of simple processes is a random variable with the following properties.

(i) (Linearity). If $X_{s}$ and $Y_{s}$ are simple processes and $\alpha$ and $\beta$ are some constants then
\[\int_{0}^{t}(\alpha X_{s}+\beta Y_{s})dW_{s}=\alpha\int_{0}^{t}X_{s}dW_{s}+\beta\int_{0}^{t}Y_{s}dW_{s}.\]
The next two properties hold if the random variables $\xi_{i}$’s are square integrable, $\mathbb{E}[\xi_{i}^{2}]<\infty$.

(ii) (Zero mean property).
\[\mathbb{E}\left [\int_{0}^{t}X_{s}dW_{s}\right ]=0\]

(iii) (Isometry property).
\[\mathbb{E}\left [\left (\int_{0}^{t}X_{s}dW_{s}\right )^{2}\right ]=\int_{0}^{t}\mathbb{E}[X_{t}^{2}]dt.\]

Proof of linearity of the integral follows from the fact that a linear combination of simple functions is again a simple function. Property (ii) follows by the martingale property of Brownian motion as in (\ref{kleeq44}). Note that provided $\xi_{i}$’s are square integrable,
\[\mathbb{E}\left [\left |\xi_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\right |\right ]\leq\sqrt{\mathbb{E}[\xi_{i}^{2}]\cdot \mathbb{E}[(W_{t_{i+1}}-W_{t_{i}})^{2}]}<\infty\]
by the Cauchy-Schwarz inequality. To prove Property (iii), we write the square as the double sum
\[\mathbb{E}\left [\left (\sum_{i=1}^{n-1}\xi_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\right )^{2}\right ]=\sum_{i=0}^{n-1}\mathbb{E}\left [\xi_{i}^{2}\cdot
(W_{t_{i+1}}-W_{t_{i}})^{2}\right ]+2\sum_{i<j}\mathbb{E}\left [\xi_{i}\xi_{j}\cdot(W_{t_{i+1}}-W_{t_{i}})\cdot (W_{t_{j+1}}-W_{t_{j}})\right ],\]
and use martingale property
\begin{align*}
\sum_{i=0}^{n-1}\mathbb{E}\left [\xi_{i}^{2}\cdot (W_{t_{i+1}}-W_{t_{i}})^{2}\right ] & =\sum_{i=0}^{n-1}\mathbb{E}\left [\mathbb{E}\left .\left [\xi_{i}^{2}\cdot
(W_{t_{i+1}}-W_{t_{i}})^{2}\right |{\cal F}_{t_{i}}\right ]\right ]\\
& =\sum_{i=0}^{n-1}\mathbb{E}\left [\xi_{i}^{2}\cdot \mathbb{E}\left .\left [(W_{t_{i+1}}-W_{t_{i}})^{2}\right |{\cal F}_{t_{i}}\right ]\right ]=
\sum_{i=0}^{n-1}\mathbb{E}[\xi_{i}^{2}\cdot (t_{i+1}-t_{i}).
\end{align*}
The last sum is exactly $\int_{0}^{t} \mathbb{E}[X_{s}^{2}]ds$ since $X_{s}=\xi_{i}$ on $(t_{i},t_{i+1}]$. By conditionsing in a similar way, we obtain for $i<j$,
\[\mathbb{E}\left [\xi_{i}\xi_{j}\cdot (W_{t_{i+1}}-W_{t_{i}})\cdot (W_{t_{j+1}}-W_{t_{j}})\right ]=0\]
Then Property (iii) is proved.

Next we discuss the Ito integral of general processes. Let $\{X^{(n)}_{s}\}_{n\in {\bf N}}$ be a sequence of simple processes convergent in probability to the process $X_{s}$. Then, under some conditions, the
sequence of their integrals $\int_{0}^{t}X_{s}^{(s)}dW_{s}$ also converges in probability. This limit is taken to be the integral $\int_{0}^{t}X_{s}dW_{s}$.

Example. We find $\int_{0}^{t}W_{s}dW_{s}$. Let $\pi_{n}=\left\{0=t_{1}^{(n)}<t_{1}^{(n)}<\cdots <t^{(n)}_{k_{n}}=t\right\}$ be a partition of $[0,t]$, and let
\[X^{(n)}_{s}=\sum_{i=0}^{k_{n}-1}W_{t^{(n)}_{i}}\cdot 1_{(t_{i}^{(n)},t_{i+1}^{(n)}]}(s).\]
Then for any $n$, $X^{(n)}_{s}$ is a simple predictable process. Here $\xi_{i}=W_{t^{(n)}_{i}}$. Take a sequence such that $\parallel\pi_{n}\parallel\rightarrow 0$ as $n\rightarrow\infty$. Then it is easy to see that for any fixed $t$, $\lim_{n\rightarrow\infty}X_{s}^{(n)}=W_{s}$ a.s. by the continuity of Brownian paths. Ito integral of this simple process is given by
\[\int_{0}^{t}X_{s}^{(n)}dW_{s}=\sum_{i=0}^{k_{n}-1}W_{t_{i}^{(n)}}\cdot\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right ).\]
We show that this sequence of integrals converges in probability and identify the limit. Adding and subtracting $W^{2}_{t_{i+1}^{(n)}}$, we obtain
\[W_{t_{i}^{(n)}}\cdot\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )=\frac{1}{2}\left (W^{2}_{t_{i+1}^{(n)}}-W^{2}_{t_{i}^{(n)}}-\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )^{2}\right ),\]
and
\begin{align*}
\int_{0}^{t}X_{s}^{(n)}dW_{s} & =\frac{1}{2}\sum_{i=1}^{k_{n}-1}\left (W^{2}_{t_{i+1}^{(n)}}-W^{2}_{t_{i}^{(n)}}\right )-\frac{1}{2}\sum_{i=0}^{k_{n}-1}\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )^{2}\\
& =\frac{1}{2}W^{2}_{t}-\frac{1}{2}W_{0}^{2}-\frac{1}{2}\sum_{i=0}^{k_{n}-1}\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )^{2}
\end{align*}
since the first term is a telescopic one. Notice that by the definition of the quadratic variation of Brownian motion the second sum converges in probability to the limit $t$. Therefore $\int_{0}^{t}X_{s}^{(n)}dW_{s}$ converges in probability, and the limit is
\[\int_{0}^{t}W_{s}dW_{s}=\lim_{n\rightarrow\infty}\int_{0}^{t}X_{s}^{(n)}dW_{s}=\frac{1}{2}W_{t}^{2}-\frac{1}{2}t. \sharp\]

The approach of defining the integral by approximation can be carried out for the class of predictable processes $\{X_{s}\}_{0\leq s\leq t}$ satisfying the condition
\[\int_{0}^{t}X_{s}^{2}ds<\infty\mbox{ a.s.}\]

Let $\{{\cal F}_{s}\}_{s\geq 0}^{W}$ be the filtration generated by the Brownian motion $W$. Then, for a ${\cal F}^{W}$-adapted process $X$, $X_{s}$ can depend on the past values of $W_{u}$ for $u\leq s$, but not on future values of $W_{u}$ for $u>s$. Intuitively, an adapted process is predictable if for any $s$, the value $X_{s}$ is determined by the values of $W_{u}$ for $u<s$. The formal definition of predictable processes is rather technical (ref. (\ref{chueq*51}), the predictable $\sigma$-filed ${\cal P}$ is generated by the adapted left-continuous processes, and a process is called predictable if it is ${\cal P}$-measurable). For our purposes it is enough to describe a subclass of predictable processes which can be defined constructively. $X$ is {\bf predictable} if it is one of the following

(a) a left-continuous adapted process;
(b) a limit (a.s. in probability) of left-continuous adapted processes;
(c) a Borel measurable function of a predictable process.

In particular any adapted and continuous process $X$ is predictable.

Definition. A simple predictable process is an adapted left-continuous step function. $\sharp$

It can be shown that under some conditions a general predictable process is a limit in probability of simple predictable processes. The Ito integral for a general predictable process is defined as a limit of integrals of
simple processes.

\begin{equation}{\label{klet43}}\tag{51}\mbox{}\end{equation}

Theorem \ref{klet43}. Let $X$ be a predictable process such that $\int_{0}^{t}X_{s}^{2}ds<\infty$ a.s. Then Ito integral $\int_{0}^{t}X_{s}dW_{s}$ is defined and satisfies Properties (i)–(iii) above.

Proof. Only outline of proof is given. Firstly the result is proved for predictable processes that satisfy an additional assumption
\begin{equation}{\label {kleeq413}}\tag{52}
\int_{0}^{t}\mathbb{E}[X_{s}^{2}]ds<\infty .
\end{equation}
For such processes there exists a sequence of simple processes $\{X_{s}^{(n)}\}_{n\in {\bf N}}$ satisfying
\begin{equation}{\label {kleeq414}}\tag{53}
\lim_{n\rightarrow\infty}\mathbb{E}\left [\int_{0}^{t}\left |X_{s}-X_{s}^{(n)}\right |^{2}\right ]ds=0.
\end{equation}
Indeed, for a continuous process $X$ satisfying (\ref{kleeq413}), such $X_{s}^{(n)}$ can be taken as
\[X_{0}+\sum_{k=0}^{2^{n}-1}X_{\frac{kt}{2^{n}}}\cdot 1_{(\frac{kt}{2^{n}},\frac{(k+1)t}{2^{n}}]}(s),\]
and (\ref{kleeq414}) is verified by the dominated convergence theorem. If $X$ is not continuous, the construction of approximating processes is more involved. Ito integrals for simple processes are defined by
(\ref{kleeq46}). It is possible to show that they form a Cauchy sequence in $L^{2}$, that is,
\[\mathbb{E}\left [\left |\int_{0}^{t}X_{s}^{(m)}dW_{s}-\int_{0}^{t}X_{s}^{(n)}dW_{s}\right |^{2}\right ]\rightarrow 0\mbox{ as }n,m\rightarrow\infty .\]
This implies, by completeness of $L^{2}$, that $\int_{0}^{t}X_{s}^{(n)}dW_{s}$ converges to a limit in $L^{2}$. It is not hard to check that this limit does not depend on the choice of the approximating sequence. The limit of $\int_{0}^{t}X_{s}^{(n)}dW_{s}$ is called $\int_{0}^{t}X_{s}dW_{s}$. Now consider predictable processes with the property that $\int_{0}^{t}X_{s}^{2}ds$ is finite with probability one, but not necessarily of finite expectation. Using the previous result, one can approximate such processes by simple processes by taking limit in probability. The sequence of corresponding Ito integral is a Cauchy sequence in probability. It converges to a limit $\int_{0}^{t}X_{s}dW_{s}$. $\blacksquare$

Corollary. If $X$ is a continuous adapted process then Ito integral $\int_{0}^{t}X_{s}dW_{s}$ exists.

Proof. Since $X$ is adapted and continuous, it is predictable. Since any path of $X$ is continuous, it is bounded on any finite time interval. Therefore $\int_{0}X_{s}^{2}ds<\infty$ a.s., and the result follows by
Theorem \ref{klet43}. $\blacksquare$

\begin{equation}{\label{klec432}}\tag{54}\mbox{}\end{equation}

Corollary \ref{klec432}. A continuous adapted process $X$ can be approximated by the following simple left-continuous processes. $X_{0}^{(n)}=X_{0}$, and for $s>0$
\[X_{s}^{(n)}=\sum_{i=0}^{k_{n}-1}X_{t_{i}^{(n)}}\cdot 1_{(t_{i}^{(n)},t_{i+1}^{(n)}]}(s),\]
where $\pi =\{t_{i}^{(n)}\}_{n\in {\bf N}}$ is a partition of $[0,t]$ with $\parallel\pi_{n}\parallel\rightarrow 0$ as $n\rightarrow\infty$. If $\int_{0}^{t}X_{s}^{2}ds<\infty$, then $\int_{0}^{t}X_{s}dW_{s}$ is approximated by
\begin{equation}{\label {kleeq416}}\tag{55}
\sum_{i=0}^{k_{n}-1}X_{t_{i}^{(n)}}\cdot\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right ),
\end{equation}
where the approximating sums $(\ref{kleeq416})$ converges in probability.

Proof. By continuity of $X$, it follows that for any $s$, $X_{s}^{(n)}\rightarrow X_{s}$ as $n\rightarrow\infty$. Since $X_{s}^{(n)}$ is a simple predictable process with Ito integral given by (\ref{kleeq416}), the result follows by Theorem \ref{klet43}. $\blacksquare$

In approximation of Stieltjes integral by sums, the function $f$ on the interval $[t_{i},t_{i+1}]$ is replaced by its value at some middle point $\theta_{i}\in [t_{i},t_{i+1}]$. In approximating Ito integral, the left-most point $t_{i}$ is taken for $\theta_{i}$. Note that when $\int_{0}^{t}X_{s}dW_{s}<\infty$ a.s. Ito integral $\int_{0}^{t}X_{s}dW_{s}$ is defined, but its mean and variance may not exist when condition
$\int_{0}^{t}\mathbb{E}[X_{s}^{2}]ds<\infty$ fails. We also note that the monotonicity $X_{s}\leq Y_{s}$ does not imply $\int_{0}^{t}X_{s}dW_{s}\leq\int_{0}^{t}Y_{s}dW_{s}$.

Let $X$ be predictable such that $\int_{0}^{t}X_{s}^{2}dW_{s}<\infty$ with probability one, so that $\int_{0}^{s}X_{u}dW_{u}$ is defined for any $s\leq t$. Since it is a random variable for any fixed $s$,
$\int_{0}^{s}X_{u}dW_{u}$ as a function of the upper limit $s$ defines a stochastic process
\[I_{s}=\int_{0}^{s}X_{u}dW_{u}.\]
It is possible to show that there is a version of Ito integral $I_{s}$ with continuous sample paths. We also see that Ito integral has infinite variation. In what follows it is always assumed that the continuous version of Ito integral is taken.

It is intuitively clear from the construction of Ito integrals that they are adapted. To see this more formally, Ito integrals of simple processes are clearly adapted, and also continuous, and therefore predictable. Since $I_{s}$ is a limit of integrals of simple processes, it is itself predictable. Suppose that in addition to condition $\int_{0}^{t}X_{s}^{2}ds<\infty$ a.s. and $\int_{0}^{t}\mathbb{E}[X_{s}^{2}]ds<\infty$ (The latter implies the former by Fubini’s theorem) Then $I_{s}=\int_{0}^{s}X_{u}dW_{u}$ for $0\leq s\leq t$, is defined and possesses first two moments. In the same way as in the proof of the zero mean property of Ito integral, it can be shown that for $t_{1}<t_{2}$,
\[\mathbb{E}\left .\left [\int_{t_{1}}^{t_{2}}X_{s}dW_{s}\right |{\cal F}_{t_{1}}\right ]=0.\]
Thus
\[\mathbb{E}\left .\left [I_{t_{2}}\right |{\cal F}_{t_{1}}\right ]=\mathbb{E}\left .\left [\int_{0}^{t_{2}}X_{u}dW_{u}\right |{\cal F}_{t_{1}}\right ]
=\int_{0}^{t_{1}}X_{u}dW_{u}+\mathbb{E}\left .\left [\int_{t_{1}}^{t_{2}}X_{u}dW_{u}\right |{\cal F}_{t_{1}}\right ]=I_{t_{1}}.\]
Therefore $\{I_{s}\}_{0\leq s\leq t}$ is a martingale. Second moment of $I_{s}$are given by the isometry property
\[\mathbb{E}\left [\left (\int_{0}^{s}X_{u}dW_{u}\right )^{2}\right ]=\int_{0}^{s}\mathbb{E}[X_{u}^{2}]du.\]
Note that $I_{s}$ is square integrable since $\sup_{s\leq t}\mathbb{E}[I_{s}^{2}]<\infty$. We formulate this as follows

\begin{equation}{\label{klet44}}\tag{56}\mbox{}\end{equation}

Proposition \ref{klet44}. Let $X$ be a predictable process such that $\int_{0}^{t}\mathbb{E}[X_{s}^{2}]ds<\infty$. Then $I_{s}=\int_{0}^{s}X_{u}dW_{u}$ for $0\leq s\leq t$ is a continuous zero mean square integrable martingale. $\sharp$

If $\int_{0}^{t}\mathbb{E}[X_{s}^{2}]ds=\infty$, then Ito integral $\int_{0}^{s}X_{u}dW_{u}$ may fail to be a martingale, but it is always a local martingale.

Corollary. For any bounded Borel measurable function $f$ on $\mathbb{R}$, $\int_{0}^{s}f(W_{u})dW_{u}$ is a square integrable martingale.

Proof. $X_{s}=f(W_{s})$ is predictable, and since $|f(x)|<c$ for some constant $c>0$, $\int_{0}^{t}\mathbb{E}[f^{2}(W_{s})]ds\leq ct$. The result follows by Proposition \ref{klet44}. $\blacksquare$

Ito’s Formula.

\begin{equation}{\label{klet47}}\tag{57}\mbox{}\end{equation}

Proposition \ref{klet47}. Let $W$ be a Brownian motion. If $g$ is a bounded continuous function and $\pi_{n}=\{t_{i}^{(n)}\}_{i=0}^{k_{n}}$ is a partition of $[0,t]$, then for any $\theta_{i}^{(n)}$ between $W_{t_{i}^{(n)}}$ and $W_{t_{i+1}^{(n)}}$, limit in probability
\[\lim_{\parallel\pi_{n}\parallel\rightarrow 0}\sum_{i=0}^{k_{n}-1}g(\theta_{i}^{(n)})\cdot \left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )^{2}=\int_{0}^{t}g(W_{s})ds. \sharp\]

From Proposition \ref{klet47}, the quadratic variation of the Brownian motion $W$ is
\[\langle W\rangle_{t}=\lim_{\parallel\pi_{n}\parallel\rightarrow 0}\sum_{i=0}^{k_{n}-1}\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )^{2}=\int_{0}^{t}ds=t.\]

Theorem. (Ito’s formula for Brownian motion). Let $W$ be a Brownian motion. If $f(x)$ is a twice continuously differentiable function, then for any $t$
\begin{equation}{\label {kleeq429}}\tag{58}
f(W_{t})=f(0)+\int_{0}^{t}f'(W_{s})dW_{s}+\frac{1}{2}\int_{0}^{t}f”(W_{s})ds.
\end{equation}

Proof. Let $\pi_{n}=\{t_{i}^{(n)}\}_{i=0}^{k_{n}}$ be a partition of $[0,t]$. Clearly,
\[f(W_{t})=f(0)+\sum_{i=o}^{k_{n}-1}\left (f(W_{t_{i+1}^{(n)}})-f(W_{t_{i}^{(n)}})\right ).\]
Apply now Taylor’s formula to $f(W_{t_{i+1}^{(n)}})-f(W_{t_{i}^{(n)}})$ to obtain
\[f(W_{t_{i+1}^{(n)}})-f(W_{t_{i}^{(n)}})=f'(W_{t_{i}^{(n)}})\cdot (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}})+\frac{1}{2}f”(\theta_{i}^{(n)}\cdot (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}})^{2},\]
where $\theta_{i}^{(n)}$ is between $W_{t_{i}^{(n)}}$ and $W_{t_{i+1}^{(n)}}$. Thus
\[f(W_{s})=f(0)+\sum_{i=0}^{k_{n}-1}f'(W_{t_{i}^{(n)}})\cdot (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}})+\frac{1}{2}\sum_{i=0}^{k_{n}-1}f”(\theta_{i}^{(n)}\cdot(W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}})^{2}.\]
Taking limits as $\parallel\pi_{n}\parallel\rightarrow 0$, the first sum converges to the Ito integral $\int_{0}^{t}f'(W_{s})dW_{s}$ (see Corollary \ref{klec432}); the second sum converges to $\int_{0}^{t}f”(W_{s})ds$ by Proposition \ref{klet43}, and the result follows. $\blacksquare$

An Ito process is an Ito integral plus an adapted absolutely continuous process of finite variation. Process $Y$ is called an Ito process if for any $0\leq s\leq t$ it can be represented as
\begin{equation}{\label {kleeq432}}\tag{59}
Y_{s}=Y_{0}+\int_{0}^{s}\mu_{u}du+\int_{0}^{t}\sigma_{u}dW_{u},
\end{equation}
where processes $\mu_{s}$ and $\sigma_{s}$ satisfy the following conditions

$\mu_{s}$ is adapted and $\int_{0}^{t}|\mu_{s}|ds<\infty$ a.s.
$\sigma_{s}$ is predictable and $\int_{0}^{t}\sigma_{s}^{2}ds<\infty$ a.s.

If $Y$ is an Ito process given by (\ref{kleeq432}) then it has a stochastic differential on $[0,t]$
\[dY_{s}=\mu_{s}ds+\sigma_{s}dW_{s}\mbox{ for }0\leq s\leq t.\]
Function $\mu$ is called the drift coefficient and function $\sigma$ is called the diffusion coefficient. For a $C^{2}$ function $f$, the Ito’s formula (\ref{kleeq429}) in differential form is given by
\[df(W_{s})=f'(W_{s})dW_{s}+\frac{1}{2}f”(W_{s})ds.\]

Example. We will find $d(e^{W_{s}})$. By using Ito’s formula with $f(x)=e^{x}$, we have
\[d(e^{W_{s}})=df(W_{s})=f'(W_{s})dW_{s}+\frac{1}{2}f”(W_{s})ds=e^{W_{s}}dW_{s}+\frac{1}{2}e^{W_{s}}ds.\]
Thus $X_{s}=e^{W_{s}}$ has stochastic differential
\[dX_{s}=X_{s}dW_{s}+\frac{1}{2}X_{s}ds. \sharp\]

Martingale Representations.

\begin{equation}{\label{klet737}}\tag{60}\mbox{}\end{equation}

Proposition \ref{klet737}. Let $W$ be a Brownian motion and a filtration $\{{\cal F}_{s}^{W}\}_{0\leq s\leq t}$ generated by $W$. If $\{M_{s}\}_{0\leq s\leq t}$ is a square integrable martingale adapted to $\{{\cal F}_{s}\}_{0\leq s\leq t}$, then there exists a predictable process $H$ such that $\mathbb{E}\left [\int_{0}^{t}H_{s}^{2}ds\right ]<\infty$ and
\[M_{s}=M_{0}+\int_{0}^{s}H_{u}dW_{u}\]
for $0\leq s\leq t$. $\sharp$

Using this result the following representation of random variable is obtained.

Proposition. Let $W$ be a Brownian motion and $Y$ be a square integrable random variable. Then there exists a predictable process $H$ such that $Y=\mathbb{E}[Y]+\int_{0}^{t} H_{s}dW_{s}$. Moreover if $Y$ ad $W$ have joint Gaussian distribution, then the process $H$ is deterministic.

Proof. Take $M_{s}=\mathbb{E}[Y|{\cal F}_{s}]$. Then $\{M_{s}\{_{0\leq s\leq t}$ is a square integrable martingale by Jensen’s inequality. Hence by Proposition \ref{klet737} there exists $H$ such that $M_{s}=M_{0}+\int_{0}^{s}H_{u}dW_{u}$. Taking $s=t$ gives the result. We omit the Gaussian part of proof. $\blacksquare$

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

Approach IV (Convergence in Probability).

This approach follows from Friedman \cite{fri}.

Definition. A $n$-dimensional stochastic process $\{{\bf X}_{t}\}_{t\in I}$ is called separable if there exists a countable sequence $\{t_{i}\}_{i\in {\bf N}}$ that is a dense subset of $I$ and a subset $N$ of $\Omega$ with $\mathbb{E}(N)=0$ such that if $\omega\not\in N$,
\[\left\{X_{t}(\omega )\in F\mbox{ for all }t\in J\right\}=\left\{X_{t_{i}}(\omega )\in F\mbox{ for all }t_{i}\in J\right\}\]

for any open subset $J$ of $I$ and for any closed subset $F$ of $\mathbb{R}^{n}$. $\sharp$

Two $n$-dimensional stochastic processes $\{{\bf X}_{t}\}_{t\in I}$ and $\{{\bf Y}_{t}\}_{t\in I}$ defined on the same probability space are said to be stochastically equivalent if
\[\mathbb{P}\left\{{\bf X}_{t}\neq {\bf Y}_{t}\right\}=0\mbox{ for all }t\in I.\]
We then say that $\{{\bf Y}_{t}\}_{t\in I}$ is a version of $\{{\bf X}_{t}\}_{t\in I}$.

Proposition. Any $n$-dimensional process is stochastically equivalent to a separable stochastic process. $\sharp$

Definition. A stochastic process $\{X_{t}\}_{\alpha\leq t\leq\beta}$ is called nonanticipative with respect to the filtration $\{{\cal F}_{t}\}_{\alpha\leq t\leq\beta}$ if

$\{X_{t}\}_{\alpha\leq t\leq\beta}$ is a separable process;
$\{X_{t}\}_{\alpha\leq t\leq\beta}$ is a measurable process, i.e., the function $(t,\omega )\mapsto X_{t}(\omega )$ from $[\alpha ,\beta ] \times\Omega$ into $\mathbb{R}$ is measurable;
$X_{t}$ is ${\cal F}_{t}$-measurable for each $t\in I$, i.e., $\{X_{t}\}_{\alpha\leq t\leq\beta}$ is adapted to $\{{\cal F}_{t}\}_{\alpha\leq t\leq\beta}$. $\sharp$

We denote by ${\cal L}^{p}([\alpha ,\beta ])$ ($1\leq p\leq\infty$) the class of all nonanticipative processes $X_{t}$ satisfying
\[\mathbb{P}\left\{\int_{\alpha}^{\beta}|X_{t}|^{p}dt<\infty\right\}=1\mbox{ if }
1\leq p<\infty\mbox{ and }\mathbb{P}\left\{\mbox{ess}\sup_{\alpha\leq t\leq\beta}|X_{t}|<\infty\right\}=1\mbox{ if }p=\infty .\]
We denote by ${\cal M}^{p}([\alpha ,\beta ])$ the subset of ${\cal L}^{p}([\alpha ,\beta ])$ consisting of all processes $X_{t}$ satisfying
\[\mathbb{E}\left [\int_{\alpha}^{\beta}|X_{t}|^{p}dt\right ]<\infty\mbox{ if }
1\leq p<\infty\mbox{ and }\mathbb{E}\left [\mbox{ess}\sup_{\alpha\leq t\leq\beta}|X_{t}\right ]|<\infty\mbox{ if }p=\infty .\]

A process $\{X_{t}\}_{\alpha\leq t\leq\beta}$ is called a simple process if there exists a partition $\alpha =t_{0}<t_{1}<\cdots <t_{n}=\beta$ of $[\alpha ,\beta ]$ such that $X_{t}=X_{t_{i}}$ if $t\in [t_{i},t_{i+1})$ for $0\leq i\leq n-1$.

\begin{equation}{\label{fril411}}\tag{61}\mbox{}\end{equation}

Proposition \ref{fril411}. Let $X\in {\cal L}^{2}([\alpha ,\beta ])$. Then, we have the following properties.

(i) There exists a sequence of continuous processes $Y^{(n)}\in {\cal L}^{2}([\alpha ,\beta ])$ satisfying
\[\lim_{n\rightarrow\infty}\int_{\alpha}^{\beta}\left |X_{t}-Y_{t}^{(n)}\right |^{2}dt=0\mbox{ a.s.};\]

(ii) There exists a sequence of simple processes $X^{(n)}\in {\cal L}^{2}([\alpha ,\beta ])$ satisfying
\[\lim_{n\rightarrow\infty}\int_{\alpha}^{\beta}\left |X_{t}-X_{t}^{(n)}\right |^{2}dt=0\mbox{ a.s.}. \sharp\]

Proposition. Let $X\in {\cal M}^{2}([\alpha ,\beta ])$. Then, we have the following properties.

(i) There exists a sequence of continuous processes $Y^{(n)}\in {\cal M}^{2}([\alpha ,\beta ])$ satisfying
\[\lim_{n\rightarrow\infty}\mathbb{E}\left [\int_{\alpha}^{\beta}\left |X_{t}-Y_{t}^{(n)}\right |^{2}dt\right ]=0.\]

(ii) There exists a sequence of bounded simple processes $X^{(n)}\in {\cal M}^{2}([\alpha ,\beta ])$ such that
\[\lim_{n\rightarrow\infty}\mathbb{E}\left [\int_{\alpha}^{\beta}\left |X_{t}-X_{t}^{(n)}\right |^{2}dt\right ]=0. \sharp\]

Let $W$ be a Brownian motion and $X_{t}$ be a simple process in ${\cal L}^{2}([\alpha ,\beta ])$, say $X_{t}=X_{i}$ if $t\in [t_{i},t_{i+1})$ for $i=0,1,\cdots ,n-1$. The random variable
\[\sum_{i=0}^{n-1}X_{t_{i}}\cdot (W_{t_{i+1}}-W_{t_{i}})=\sum_{i=0}^{n-1}X_{i}\cdot (W_{t_{i+1}}-W_{t_{i}})\]
is denoted by $\int_{\alpha}^{\beta}X_{t}dW_{t}$ is called the stochastic integral of $f$ with respect to the Brownian motion $W$; it is also called the Ito integral.

Proposition. We have the following properties.

(i) Let $X$ and $Y$ be two simple processes in ${\cal L}^{2}([\alpha ,\beta ])$ and $c_{1}$ and $c_{2}$ be two real numbers. Then $c_{1}X+c_{2}Y$ is in ${\cal L}^{2}([\alpha ,\beta ])$ and
\[\int_{\alpha}^{\beta}\left (c_{1}X_{t}+c_{2}Y_{t}\right )dW_{t}=c_{1}\int_{\alpha}^{\beta}X_{t}dW_{t}+c_{2}\int_{\alpha}^{\beta}Y_{t}dW_{t}.\]

(ii) If $X$ is a simple process in ${\cal M}^{2}([\alpha ,\beta ])$, then
\[\mathbb{E}\left [\int_{\alpha}^{\beta}X_{t}dW_{t}\right ]=0\mbox{ and }\mathbb{E}\left [\left |\int_{\alpha}^{\beta}X_{t}dW_{t}\right |^{2}\right ]=
\mathbb{E}\left [\int_{\alpha}^{\beta}X_{t}^{2}dt\right ]. \sharp\]

\begin{equation}{\label{fril423}}\tag{62}\mbox{}\end{equation}

Proposition \ref{fril423}. For any simple process $X$ in ${\cal L}^{2}([\alpha ,\beta ])$ and for any $\epsilon >0$, $N>0$, we have
\[\mathbb{P}\left\{\left |\int_{\alpha}^{\beta}X_{t}dW_{t}\right |>\epsilon\right\}
\leq \frac{N}{\epsilon^{2}}+\mathbb{P}\left\{\int_{\alpha}^{\beta}X_{t}^{2}dt>N\right\}. \sharp\]

We shall now proceed to define the stochastic integral for any $X$ in ${\cal L}^{2}([\alpha ,\beta ])$. By Proposition~\ref{fril411}, there is a sequence of simple processes $X^{(n)}$ in ${\cal L}^{2}([\alpha ,\beta ])$ satisfying
\[\int_{\alpha}^{\beta}\left |X_{t}-X_{t}^{(n)}\right |^{2}dt\rightarrow 0\mbox{ in probability as }n\rightarrow\infty .\]
Hence
\[\int_{\alpha}^{\beta}\left |X_{t}^{(n)}-X_{t}^{(m)}\right |^{2}dt\rightarrow 0\mbox{ in probability as }n,m\rightarrow\infty .\]
By Proposition \ref{fril423}, for any $\epsilon >0$, $\rho >0$,
\[\mathbb{P}\left\{\left |\int_{\alpha}^{\beta}X_{t}^{(n)}dW_{t}-\int_{\alpha}^{\beta}X_{t}^{(m)}dW_{t}\right |>\epsilon\right\}\leq\rho +\mathbb{P}\left\{
\int_{\alpha}^{\beta}\left |X_{t}^{(n)}-X_{t}^{(m)}\right |^{2}>\epsilon^{2}\rho\right\}.\]
It follows that the sequence
\[\left\{\int_{\alpha}^{\beta}X_{t}^{(n)}dW_{t}\right\}_{n\in {\bf N}}\]
is convergent in probability. We denote the limit by
\[\int_{\alpha}^{\beta}X_{t}dW_{t}\]
and call the stochastic integral of $X_{t}$ with respect to the Brownian motion $W_{t}$.

The above definition is independent of the particular sequence $\{X^{(n)}\}_{n\in {\bf N}}$. For if $\{Y^{(n)}\}_{n\in {\bf N}}$ is another sequence of simple processes in ${\cal L}^{2}([\alpha ,\beta ])$ converging to $X$ in the sense that
\[\int_{\alpha}^{\beta}\left |X_{t}-Y_{t}^{(n)}\right |^{2}dt\rightarrow 0\mbox{ in probability as }n\rightarrow\infty ,\]
then the sequence $\{Z^{(n)}\}_{n\in {\bf N}}$ defined by $Z^{(2n)}=X^{(n)}$ and $Z^{(2n+1)}=Y^{(n)}$ is also convergent to $X$ in the same sense. But then, by what we have proved, the sequence
$\left\{\int_{\alpha}^{\beta}Z^{(n)}dW_{t}\right\}_{n\in {\bf N}}$ is convergent in probability. It follows that the limits (in probability) of $\int_{\alpha}^{\beta}X^{(n)}dW_{t}$ and $\int_{\alpha}^{\beta}Y^{(n)}dW_{t}$ are equal a.s.

\begin{equation}{\label{frit425}}\tag{63}\mbox{}\end{equation}

Proposition \ref{frit425}. We have the following properties.

(i) Let $X$ and $Y$ be two processes in ${\cal L}^{2}([\alpha ,\beta ])$ and $c_{1}$ and $c_{2}$ be two real numbers. Then $c_{1}X+c_{2}Y$ is in ${\cal L}^{2}([\alpha ,\beta ])$ and
\[\int_{\alpha}^{\beta}\left (c_{1}X_{t}+c_{2}Y_{t}\right )dW_{t}=c_{1}\int_{\alpha}^{\beta}X_{t}dW_{t}+c_{2}\int_{\alpha}^{\beta}Y_{t}dW_{t}.\]

(ii) If $X$ is a process in ${\cal M}^{2}([\alpha ,\beta ])$, then
\[\mathbb{E}\left [\int_{\alpha}^{\beta}X_{t}dW_{t}\right ]=0\mbox{ and }\mathbb{E}\left [\left |\int_{\alpha}^{\beta}X_{t}dW_{t}\right |^{2}\right ]=
\mathbb{E}\left [\int_{\alpha}^{\beta}X_{t}^{2}dt\right ].\]

(iii) If $X$ is a process in ${\cal L}^{2}([\alpha ,\beta ])$ and for any $\epsilon >0$, $N>0$, we have
\[\mathbb{P}\left\{\left |\int_{\alpha}^{\beta}X_{t}dW_{t}\right |>\epsilon\right\}\leq \frac{N}{\epsilon^{2}}+\mathbb{P}\left\{\int_{\alpha}^{\beta}X_{t}^{2}dt>N\right\}. \sharp\]

Proposition. Let $X,X^{(n)}$ be in ${\cal L}^{2}([\alpha ,\beta ])$ and suppose that
\[\int_{\alpha}^{\beta}\left |X_{t}-X_{t}^{(n)}\right |^{2}dt\rightarrow 0\mbox{ in probability as }n\rightarrow\infty .\]
Then
\[\int_{\alpha}^{\beta}X_{t}^{(n)}dW_{t}\rightarrow\int_{\alpha}^{\beta}X_{t}dW_{t}\mbox{ in probability as }n\rightarrow\infty . \sharp\]

Proposition. Let $X\in {\cal M}^{2}([\alpha ,\beta ])$. Then
\[\mathbb{E}\left .\left [\int_{\alpha}^{\beta}X_{t}dW_{t}\right |{\cal F}_{\alpha}\right ]=0\]
and
\[\mathbb{E}\left .\left [\left |\int_{\alpha}^{\beta}X_{t}dW_{t}\right |^{2}\right |{\cal F}_{\alpha}\right ]=\mathbb{E}\left .\left [\int_{\alpha}^{\beta}
X_{t}^{2}dt\right |{\cal F}_{\alpha}\right ]=\int_{\alpha}^{\beta}\mathbb{E}\left .\left [X_{t}^{2}\right |{\cal F}_{\alpha}\right ]dt. \sharp\]

Proposition. If $X\in {\cal L}^{2}([\alpha ,\beta ])$ and $X$ is continuous, then, for any sequence $\{\pi_{n}\}_{n\in {\bf N}}$ of partitions $\alpha =t_{0}^{(n)}<t_{1}^{(n)}<\cdots t_{k_{n}}^{(n)}=\beta$ of $[\alpha ,\beta ]$ with mesh $\parallel\pi_{n}\parallel\rightarrow 0$,
\[\sum_{i=0}^{k_{n}-1}X_{t_{i}^{(n)}}\cdot\left (W_{t_{i+1}^{(n)}}-W_{t_{i}^{(n)}}\right )\rightarrow\int_{\alpha}^{\beta}X_{t}dW_{t}\mbox{ in probability as }n\rightarrow\infty . \sharp\]

Now we consider the indefinite integral. Let $X\in {\cal L}^{2}([0,\eta ])$ and consider the integral
\begin{equation}{\label {frieq431}}\tag{64}
I_{t}=\int_{0}^{t}X_{s}dW_{s}\mbox{ for }0\leq t\leq\eta
\end{equation}
where, by definition, $\int_{0}^{0}X_{s}dW_{s}=0$. We refer to $I_{t}$ as the indefinite integral of $X$. Notice that $I_{t}$ is ${\cal F}_{t}$-measurable.

If $X$ is a simple process, then clearly
\begin{equation}{\label {frieq432}}\tag{65}
\int_{\alpha}^{\beta}X_{s}dW_{s}+\int_{\beta}^{\gamma}X_{s}dW_{s}=\int_{\alpha}^{\gamma}X_{s}dW_{s}\mbox{ if }0\leq\alpha <\beta <\gamma\leq\eta .
\end{equation}
By approximation we find that (\ref{frieq432}) holds for any $X$ in ${\cal L}^{2}([0,\eta ])$.

Proposition. (Friedman \cite{fri}). If $X\in {\cal M}^{2}([0,\eta ])$, then the indefinite integral $\{I_{t}\}_{0\leq t\leq\eta}$ is a martingale. $\sharp$

Proposition. If $X\in {\cal L}^{2}([0,\eta ])$, then the indefinite integral $\{I_{t}\}_{0\leq t\leq\eta}$ has a continuous version. $\sharp$

From now on, when we speak of the indefinite integral $I_{t}$ of $X\in {\cal L}^{2}([0,\eta ])$, we always mean a continuous version of it.

Proposition. Let $X\in {\cal L}^{2}([0,\eta ])$. Then for any $\epsilon >0$, $N>0$, we have
\[\mathbb{P}\left\{\sup_{0\leq t\leq\eta}\left |\int_{0}^{t}X_{s}dW_{s}\right |>\epsilon\right\}\leq\frac{N}{\epsilon^{2}}+\mathbb{P}\left\{\int_{0}^{\eta}X_{t}^{2}dt>N\right\}.\]

Proposition. Let $X,X^{(n)}$ be in ${\cal L}^{2}([0,\eta ])$ and assume that
\[\int_{0}^{\eta}\left |X_{t}-X_{t}^{(n)}\right |^{2}dt\rightarrow 0\mbox{ in probability as }n\rightarrow\infty .\]
Then
\[\sup_{0\leq t\leq\eta}\left |\int_{0}^{t}X_{s}^{(n)}dW_{s}-\int_{0}^{t}X_{s}dW_{s}\right |\rightarrow 0\mbox{ in probability as }n\rightarrow\infty .\]

Proposition. Let $X\in {\cal M}^{2}([0,\eta ])$. Then
\[\mathbb{E}\left [\sup_{0\leq t\leq\eta}\left |\int_{0}^{t}X_{s}dW_{s}\right |^{2}\right ]\leq 4\mathbb{E}\left [\left |\int_{0}^{\eta}X_{t}dW_{t}\right |^{2}\right ]=
4\mathbb{E}\left [\int_{0}^{\eta}X_{t}^{2}dt\right ]. \sharp\]

It is clear that all the results derived so far regarding the definite integral of (\ref{frieq431}) can extend to definite integrals
\[\int_{\alpha}^{t}X_{s}dW_{s}.\]

Proposition. Let $X\in {\cal M}^{2}([\alpha ,\beta ])$. Then
\[\mathbb{E}\left .\left [\sup_{\alpha\leq t\leq\beta}\left |\int_{\alpha}^{t}X_{s}dW_{s}\right |^{2}\right |{\cal F}_{\alpha}\right ]\leq 4\mathbb{E}\left .\left [
\int_{\alpha}^{\beta}X_{t}^{2}dW_{t}\right |{\cal F}_{\alpha}\right ]. \sharp\]

Let $T$ be a random variable, $0\leq T\leq\eta$. If $X\in {\cal L}^{2}([0,\eta ])$, we define
\[\int_{0}^{T}X_{s}dW_{s}\]
to be the random variable $I_{T(\omega )}$, where $I_{t}$ is given by (\ref{frieq431}).

Proposition. If $X\in {\cal M}^{2}([0,T])$ and $T$ is a stopping time with respect to ${\cal F}_{t}$, $0\leq T\leq\eta$, then the process
\[\left\{\int_{0}^{T\wedge t}X_{s}dW_{s}\right\}_{0\leq t\leq\eta}\]
is a martingale and
\[\mathbb{E}\left [\int_{0}^{T\wedge t}X_{s}dW_{s}\right ]=0. \sharp\]

If $X\in {\cal L}^{2}([\alpha ,\eta ])$ for all $\eta >0$, then we say that $X$ belongs to ${\cal L}^{2}([\alpha ,\infty ))$. Similarly we define the class ${\cal M}^{2}([\alpha ,\infty ))$. Let $X\in {\cal L}^{2}([0,\eta ])$ and let $T_{1},T_{2}$ be random variables, $0\leq T_{1}\leq T_{2}\leq\eta$. We define
\[\int_{T_{1}}^{T_{2}}X_{s}dW_{s}=\int_{0}^{T_{2}}X_{s}dW_{s}-\int_{0}^{T_{1}}X_{s}dW_{s}.\]

Proposition. Let $X\in {\cal M}^{2}([0,\eta ])$ and let $T_{1},T_{2}$ be stopping times, $0\leq T_{1}\leq T_{2}\leq\eta$. Then
\[\mathbb{E}\left [\int_{T_{1}}^{T_{2}}X_{s}dW_{s}\right ]=0\mbox{ and }\mathbb{E}\left [\left |\int_{T_{1}}^{T_{2}}X_{s}dW_{s}\right |^{2}\right ]=\mathbb{E}\left [\int_{T_{1}}^{T_{2}}X_{t}^{2}dt\right ]. \sharp\]

Recall that if $T$ is a stopping time with respect to the filtration ${\cal F}_{t}$, then ${\cal F}_{T}$ denotes the $\sigma$-field of all events $A$ such that $A\cap\{T\leq s\}$ is in ${\cal F}_{s}$ for all $s\geq 0$.

Proposition. Let $X\in {\cal M}^{2}([0,\eta ])$ and let $T_{1},T_{2}$ be stopping times, $0\leq T_{1}\leq T_{2}\leq\eta$. Then
\[\mathbb{E}\left .\left [\int_{T_{1}}^{T_{2}}X_{s}dW_{s}\right |{\cal F}_{T_{1}}\right ]=0\mbox{ and }
\mathbb{E}\left .\left [\left |\int_{T_{1}}^{T_{2}}X_{s}dW_{s}\right |^{2}\right |
{\cal F}_{T_{1}}\right ]=\mathbb{E}\left .\left [\int_{T_{1}}^{T_{2}}X_{t}^{2}dt\right |{\cal F}_{T_{1}}\right ].\]}

\begin{equation}{\label{frit445}}\tag{66}\mbox{}\end{equation}

Proposition \ref{frit445}. Let $X\in {\cal L}^{2}([0,\infty ))$ and assume that
\[\int_{0}^{\infty}X_{t}^{2}dt=\infty\mbox{ with probability one.}\]
Let
\begin{equation}{\label {frieq*1}}\tag{67}
T(t)=\inf\left\{s:\int_{0}^{s}X_{u}^{2}du=t\right\},
\end{equation}
then the process
\[\widehat{W}_{t}=\int_{0}^{T(t)}X_{s}dW_{s}\]
is a Brownain motion. $\sharp$

Let
\[T^{*}(t)=\int_{0}^{t}X_{s}^{2}ds.\]
Then $T(t)$ in (\ref{frieq*1}) is the left-continuous inverse of $T^{*}$, i.e., $T(t)=\min\{s,T^{*}(s)=t\}$. $T^{*}$ is called the intrinsic time for $I_{t}=\int_{0}^{t}X_{s}dW_{s}$. Proposition \ref{frit445} asserts that there is a Brownian motion $\widehat{W}_{t}$ satisfying $\widehat{W}_{T^{*}(t)}=I_{t}$.

Ito’s Formula.

Let $\{X_{t}\}_{0\leq t\leq\eta}$ be a process such that for any $0\leq t_{1}<t_{2}\leq\eta$,
\[X_{t_{2}}-X_{t_{1}}=\int_{t_{1}}^{t_{2}}\mu_{t}dt+\int_{t_{1}}^{t_{2}}\sigma_{t}dW_{t}\]
where $\mu\in {\cal L}^{1}([0,\eta ])$ and $\sigma\in {\cal L}^{2}([0,\eta ])$. Then we say that $X_{t}$ has stochastic differential $dX_{t}$ on $[0,\eta ]$, given by
\[dX_{t}=\mu_{t}dt+\sigma_{t}dW_{t}.\]
Observe that $X_{t}$ is a nonanticipative process. It is also a continuous process. Hence, in particular, it belongs to ${\cal L}^{\infty}([0,\eta ])$. Let $Y\in {\cal L}^{\infty}([0,\eta ])$. We define
\[Y_{t}dX_{t}=Y_{t}\mu_{t}dt+Y_{t}\sigma_{t}dW_{t}.\]

Proposition. If $dX_{t}^{(i)}=\mu_{t}^{(i)}dt+\sigma_{t}^{(i)}dW_{t}$, $i=1,2$, then
\[d(X_{t}^{(1)}\cdot X_{t}^{(2)})=X_{t}^{(1)}dX_{t}^{(2)}+X_{t}^{(2)}dX_{t}^{(1)}+\sigma_{t}^{(1)}\cdot\sigma_{t}^{(2)}dt.\]
The integral form asserts that, for any $0\leq t_{1}<t_{2}\leq\eta$,
\begin{align*}
X_{t_{2}}^{(1)}X_{t_{2}}^{(2)}-X_{t_{1}}^{(1)}X_{t_{1}}^{(2)} & =\int_{t_{1}}^{t_{2}}X_{t}^{(1)}\mu_{t}^{(2)}dt+\int_{t_{1}}^{t_{2}}
X_{t}^{(1)}\sigma_{t}^{(2)}dW_{t}+\int_{t_{1}}^{t_{2}}X_{t}^{(2)}\mu_{t}^{(1)}dt+\int_{t_{1}}^{t_{2}}X_{t}^{(2)}\sigma_{t}^{(1)}dW_{t}\\
& +\int_{t_{1}}^{t_{2}}\sigma_{t}^{(1)}\sigma_{t}^{(2)}dt.
\end{align*}

Theorem. (Ito’s Formula). Let $dX_{t}=\mu_{t}dt+\sigma_{t}dW_{t}$, and let $f(x,t)$ be a continuous function in $(x,t)\in \mathbb{R}\times [0,\infty )$ together with its derivatives $\partial f/\partial x$, $\partial^{2}f/\partial x^{2}$, $\partial f/\partial t$. Then the process $f(X_{t},t)$ has a stochastic differential given by
\[df(X_{t},t)=\left (\frac{\partial f}{\partial t}(X_{t},t)+\frac{\partial f}{\partial x}(X_{t},t)+\frac{1}{2}\sigma_{t}^{2}\cdot
\frac{\partial^{2}f}{\partial x^{2}}(X_{t},t)\right )dt+\sigma_{t}\cdot\frac{\partial f}{\partial x}(X_{t},t)dW_{t}. \sharp\]

Notice that if $W_{t}$ were continuously differentiable in $t$, then the term $\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}\sigma^{2}dt$ would not appear.

Theorem. (Ito’s Formula). Let $dX_{t}^{(i)}=\mu_{t}^{(i)}dt+\sigma_{t}^{(i)}dW_{t}$, $1\leq i\leq m$, and let $f(x_{1},\cdots ,x_{m},t)$ be a continuous function in $({\bf x},t)\in \mathbb{R}^{m}\times [0,\infty )$ together with its $t$-derivative and second ${\bf x}$-derivative. Then $f(X_{t}^{(1)},\cdots ,X_{t}^{(m)},t)$ has a stochastic differential given by
\begin{align*}
df({\bf X}_{t},t) & =\left (\frac{\partial f}{\partial t}({\bf X}_{t},t)+\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}({\bf X}_{t},t)\mu_{t}^{(i)}+
\frac{1}{2}\sum_{i,j=1}^{m}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf X}_{t},t)\cdot\sigma_{t}^{(i)}\cdot\sigma_{t}^{(j)}\right )dt\\
& +\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}({\bf X}_{t},t)\cdot\sigma_{t}^{(i)}dW_{t}. \sharp
\end{align*}

Proposition. Let $X\in {\cal L}^{2}([0,\eta ])$, and let $h,k$ be any positive numbers. Then
\begin{equation}{\label {frieq468}}\tag{68}
\mathbb{P}\left\{\sup_{0\leq t\leq\eta}\left (\int_{0}^{t}X_{s}dW_{s}-\frac{h}{2}\int_{0}^{t}X_{s}^{2}ds\right )>k\right\}\leq e^{-hk}. \sharp
\end{equation}

The inequality (\ref{frieq468}) will be referred to as the exponential martingale inequality.

Corollary. If $X\in {\cal L}^{2}([0,\eta ])$, then the process
\[Y_{t}=\ex\mathbb{P}\left (\int_{0}^{t}X_{s}dW_{s}-\frac{1}{2}\int_{0}^{t}X_{s}^{2}ds\right )\]
is a supermartingale. $\sharp$

Next we discuss the $n$-dimensional stocahstic integrals. Let ${\bf W}_{t}=(W_{t}^{(1)},\cdot ,W_{t}^{(n)})$ be an $n$-dimensional Brownian motion. Let ${\cal F}_{t}$ be a filtration such that ${\bf W}_{t}$ is ${\cal F}_{t}$-measurable. We shall say that a matrix of processes belongs to ${\cal L}^{p}([\alpha ,\beta ])$ (or to ${\cal M}^{p}([\alpha ,\beta ])$) if each of its elements belongs to ${\cal L}^{p}([\alpha ,\beta ])$ (or to ${\cal M}^{p}([\alpha ,\beta ])$).

Let $\boldsymbol{\sigma}$ be an $m\times n$ matrix that belongs to ${\cal L}^{2}([\alpha ,\beta ])$. The stochastic integral $\int_{\alpha}^{\beta}\boldsymbol{\sigma}_{t}d{\bf W}_{t}$ is an $m$-vector defined by
\[\int_{\alpha}^{\beta}\boldsymbol{\sigma}_{t}d{\bf W}_{t}=\left (\sum_{j=1}^{n}\int_{\alpha}^{\beta}\sigma_{t}^{(1j)}dW_{t}^{(j)},\cdots ,
\sum_{j=1}^{n}\int_{\alpha}^{\beta}\sigma_{t}^{(ij)}dW_{t}^{(j)},\cdots ,\sum_{j=1}^{n}\int_{\alpha}^{\beta}\sigma_{t}^{(mj)}dW_{t}^{(j)}\right ).\]

If we substiute $a=\int_{t_{1}}^{t_{2}}XdW^{(i)}$ and $b=\int_{t_{1}}^{t_{2}}YdW^{(i)}$ in the identity $4ab=(a+b)^{2}-(a-b)^{2}$, and use Proposition \ref{frit425} (ii), we find that
\begin{equation}{\label {frieq471}}\tag{69}
\mathbb{E}\left [\int_{t_{1}}^{t_{2}}X_{t}dW_{t}^{(i)}\cdot\int_{t_{1}}^{t_{2}}Y_{t}dW_{t}^{(i)}\right ]=\mathbb{E}\left [\int_{t_{1}}^{t_{2}}X_{t}Y_{t}dt\right ]
\end{equation}
provided that $X$ and $Y$ belong to ${\cal M}^{2}([t_{1},t_{2}])$. We also have
\begin{equation}{\label {frieq472}}\tag{70}
\mathbb{E}\left [\int_{t_{1}}^{t_{2}}X_{t}dW_{t}^{(i)}\cdot\int_{t_{1}}^{t_{2}}Y_{t}dW_{t}^{(j)}\right ]=0\mbox{ if }i\neq j,
\end{equation}
since the integrals are independent and with zero expectation. Using (\ref{frieq471}) and (\ref{frieq472}) we easily see that if $\boldsymbol{\sigma}$ is an $m\times n$ matrix in ${\cal M}^{2}([t_{1},t_{2}])$, then
\[\mathbb{E}\left [\left |\!\left |\int_{t_{1}}^{t_{2}}\boldsymbol{\sigma}_{t}d{\bf W}_{t}\right |\!\right |^{2}\right ]=\mathbb{E}\left [\int_{t_{1}}^{t_{2}}\parallel\boldsymbol{\sigma}_{t}\parallel^{2}dt\right ]\]
where
\[\parallel\boldsymbol{\sigma}\parallel^{2}=\sum_{i=1}^{m}\sum_{j=1}^{n}(\sigma^{(ij)})^{2}.\]

Let $\{{\bf X}_{t}\}_{0\leq t\leq\eta}$ be an $n$-dimensional process, and suppose that, for any $0\leq t_{1}\leq t_{2}\leq\eta$,
\[{\bf X}_{t_{2}}-{\bf X}_{t_{1}}=\int_{t_{1}}^{t_{2}}\boldsymbol{\mu}_{t}dt+\int_{t_{1}}^{t_{2}}\boldsymbol{\sigma}_{t}d{\bf W}_{t}\]
where the $m$-vector $\boldsymbol{\mu}$ and the $m\times n$ matrix $\boldsymbol{\sigma}$ belong to ${\cal L}^{1}([0,\eta ])$ and ${\cal L}^{2}([0,\eta ])$, respectively. Then we say that ${\bf X}_{t}$ has a stochastic differential $d{\bf X}_{t}$ given by
\[d{\bf X}_{t}=\boldsymbol{\mu}_{t}dt+\boldsymbol{\sigma}_{t}d{\bf W}_{t}.\]

Theorem. (Ito’s Formula). Let $f({\bf x},t)$ be a continuous function in $({\bf x},t)\in \mathbb{R}^{m} \times [0,\infty )$ together with its derivatives $\partial f/\partial t$, $\partial f/\partial x_{i}$ and $\partial^{2}f/\partial x_{i}\partial x_{j}$. Let ${\bf X}_{t}$ be an $m$-dimensional process having a stochastic differential $d{\bf X}_{t}=\boldsymbol{\mu}_{t}dt+ \boldsymbol{\sigma}_{t}d{\bf W}_{t}$ where the $m$-vector $\boldsymbol{\mu}$ and the $m\times n$ matrix $\boldsymbol{\sigma}$ belong to ${\cal L}^{1}([0,\eta ])$ and ${\cal L}^{2}([0,\eta ])$, respectively. Then $f({\bf X}_{t},t)$ has a stochastic differential
\begin{align*}
df({\bf X}_{t},t) & =\left (\frac{\partial f}{\partial t}({\bf X}_{t},t)+\sum_{i=1}^{m}\mu_{t}^{(i)}\cdot\frac{\partial f}{\partial x_{i}}({\bf X}_{t},t)+\frac{1}{2}\sum_{l=1}^{n}\sum_{i,j=1}^{m}
\sigma_{t}^{(il)}\cdot\sigma_{t}^{(jl)}\cdot\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf X}_{t},t)\right )dt\\
& +\sum_{l=1}^{n}\sum_{j=1}^{m}\sigma_{t}^{(il)}\cdot\frac{\partial f}{\partial x_{i}}({\bf X}_{t},t)dW_{t}^{(l)}.
\end{align*}

Let us define formally the multiplication rule
\[dW^{(i)}dt=dtdW^{(i)}=dtdt=0\mbox{ and }dW^{(i)}dW^{(j)}=\delta_{ij}\cdot t\]
so that
\[dX^{(i)}dX^{(j)}=\sum_{l=1}^{n}\sigma^{(il)}\cdot\sigma^{(jl)}dt.\]
Then the Ito’s formula takes the form
\[df({\bf X}_{t},t)=\frac{\partial f}{\partial t}({\bf X}_{t},t)dt+\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}({\bf X}_{t},t)dX^{(i)}+
\frac{1}{2}\sum_{i,j=1}^{m}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf X}_{t},t)dX^{(i)}dX^{(j)}.\]

Proposition. Let ${\bf X}\in {\cal L}^{2}([0,\infty ))$ and suppose that
\[\int_{0}^{\infty}\parallel X_{t}\parallel^{2}dt=\infty\mbox{ with probability one.}\]
and define
\[T(t)=\inf\left\{s:\int_{0}^{s}\parallel X_{u}\parallel^{2}du=t\right\}.\]
Then the process
\[\widehat{W}_{t}=\int_{0}^{T(t)}{\bf X}_{s}d{\bf W}_{s}\]
is a Brownian motion. $\sharp$

Proposition. Let ${\bf X}\in {\cal }^{2}([0,\eta ])$, and let $h,k$ be any positive numbers. Then
\begin{equation}{\label {frieq4712}}\tag{71}
\mathbb{P}\left\{\sup_{0\leq t\leq\eta}\left (\int_{0}^{t}{\bf X}_{s}d{\bf W}_{s}-\frac{h}{2}\int_{0}^{t}\parallel {\bf X}_{s}\parallel^{2}ds\right )>k\right\}\leq e^{-hk}.
\end{equation}
\end{Pro}

The inequality (\ref{frieq4712}) will be referred to as the exponential martingale inequalit.

Corollary. If ${\bf X}\in {\cal L}^{2}([0,\eta ])$, then the process
\[Y_{t}=\ex\mathbb{P}\left (\int_{0}^{t}{\bf X}_{s}d{\bf W}_{s}-\frac{1}{2}\int_{0}^{t}\parallel {\bf X}_{s}\parallel^{2}ds\right )\]
is a supermartingale. $\sharp$

Convergence in L^{2}

Ito’s Formula.

Stratonovich Stochastic Integrals.

Approach II (Convergence in L^{2}).

Ito’s Formula.

Approach III (Convergence in Probability)

Ito’s Formula.

Martingale Representations.

Approach IV (Convergence in Probability).

Ito’s Formula.

Hsien-Chung Wu

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Convergence in L^{2}

Ito’s Formula.

Stratonovich Stochastic Integrals.

Approach II (Convergence in L^{2}).

Ito’s Formula.

Approach III (Convergence in Probability)

Ito’s Formula.

Martingale Representations.

Approach IV (Convergence in Probability).

Ito’s Formula.

Hsien-Chung Wu

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Stochastic Calculus for Local Martingales

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