Edwin Longsden Long (1829-1891) was an English painter.
The topics are
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Approach I.
Stochastic Integrals for Martingales.
This approach follows from Revuz and Yor \cite{rev}. Recall the definition of \({\cal H}_{c}^{2}\) in Definition \ref{revd*3}.
Definition. If \(M\in {\cal H}_{c}^{2}\), we call \(L^{2}(M)\) the space of progressively measurable processes \(X\) such that
\[\parallel X\parallel_{M}^{2}=E\left [\int_{0}^{\infty}X_{s}^{2}d\langle M\rangle_{s}\right ]<\infty. \sharp\]
For any \(A\in {\cal B}\times {\cal F}_{\infty}\), we set
\[P_{M}(A)=E\left [\int_{0}^{\infty} 1_{A}(s,\omega )d\langle M\rangle_{s}(\omega )\right ].\]
Then \(P_{M}\) is a bounded measure on \({\cal B}\times {\cal F}_{\infty}\) and the space \(L^{2}(M)\) is nothing else than the space of \(P_{M}\)-square integrable, progressively measurable functions. As usual, \({\cal L}^{2}(M)\) will denote the space of equivalence classes of elements of \(L^{2}(M)\); it is of course a Hilbert space for the norm \(\parallel\cdot\parallel_{M}\). Since those are the processes we are going to integrate, it is worth recalling that they include all the bounded and left- or right-continuous adapted processes, and, in particular, the bounded continuous adapted processes.
\begin{equation}{\label{revt422}}\tag{1}\mbox{}\end{equation}
Theorem \ref{revt422}. (Revuz and Yor \cite{rev}). Let \(M\in {\cal H}_{c}^{2}\); for each \(X\in {\cal L}^{2}(M)\), there is a unique element of \({\cal H}_{0}^{2}\), denoted by \(X\bullet M\), such that
\[\langle X\bullet M,N\rangle =X\bullet\langle M,N\rangle\]
for every \(N\in {\cal H}_{c}^{2}\), where \(X\bullet\langle M,N\rangle\) is a Stieltjes integral defined in \((\ref{reveq*16})\). The map \(X\mapsto X\bullet M\) is an isometry from \({\cal L}^{2}(M)\) into \({\cal H}_{0}^{2}\). \(\sharp\)
If \(N\in {\cal H}_{c}^{2}\) instead of \({\cal H}_{0}^{2}\), then we still have \(\langle X\bullet M,N\rangle =X\bullet\langle M,N\rangle\) because the bracket of a martingale with a constant martingale is zero.
Definition. The martingale \(X\bullet M\) is called the stochastic integral of \(X\) with respect to \(M\) and is also denoted by
\[(X\bullet M)_{t}=\int_{0}^{t}X_{s}dM_{s}. \sharp\]
We stress the fact that the stochastic integral \(X\bullet M\in {\cal H}_{0}^{2}\) vanishing at \(0\). The reason we calling \(X\bullet M\) a stochastic integral will become clearer in the sequel. We shall denote by \({\cal S}\) the space of simple processes that is the processes which can be written as
\[X=\xi_{-1}\cdot 1_{\{0\}}+\sum_{i}\xi_{i}\cdot 1_{(t_{i},t_{i+1}]}\]
where \(0=t_{0}<t_{1}<t_{2}<\cdots\), \(\lim_{i}t_{i}=\infty\), and the random variables \(\xi_{i}\) are \({\cal F}_{t_{i}}\)-measurable and uniformly bounded and \(\xi_{-1}\) is \({\cal F}_{0}\)-measurable. The space \({\cal S}\) is contained in \({\cal L}^{2}(M)\). For \(X\in {\cal S}\), we define the stochastic integral \(X\bullet M\) by
\[(X\bullet M)_{t}=\sum_{i=0}^{n-1}\xi_{i}(M_{t_{i+1}}-M_{t_{i}})+\xi_{n}(M_{t}-M_{t_{n}})\]
whenever \(t_{n}\leq t<t_{n+1}\). It is easily seen that \(X\bullet M\in {\cal H}_{0}^{2}\); moreover, considering partitions \(\pi\) including the \(t_{i}\)’s, it can be proved using the definition of the brackets, that
for any \(N\in {\cal H}_{c}^{2}\), we have \(\langle X\bullet M,N\rangle=X\bullet\langle M,N\rangle\).
\begin{equation}{\label{revp424}}\tag{2}\mbox{}\end{equation}
Proposition \ref{revp424}. (Revuz and Yor \cite{rev}). If \(X\in {\cal L}^{2}(M)\) and \(Y\in {\cal L}^{2}(K\bullet M)\) then \(XY\in {\cal L}^{2}(M)\) and \((XY)\bullet M=Y\bullet (X\bullet M)\).
Proof. Since
\[\langle X\bullet M,X\bullet M\rangle =X\bullet\langle M,X\bullet M\rangle =X\bullet (X\bullet\langle M,M\rangle )=X^{2}\bullet\langle M,M\rangle,\]
it is clear that \(XY\in {\cal L}^{2}(M)\). For \(N\in {\cal H}_{c}^{2}\), we further have
\[\langle (XY)\bullet M,N\rangle =XY\bullet\langle M,N\rangle =Y\bullet (X\bullet\langle M,N\rangle )\]
because of the obvious associativity of Stieltjes integrals, and this is equal to
\[Y\bullet\langle X\bullet M,N\rangle =\langle Y\bullet (X\bullet M),N\rangle ;\]
the uniqueness in Theorem~\ref{revt422} completes the proof. \(\blacksquare\)
Stochastic Integrals for Local Martingales.
The next result shows how stochastic integration behaves with respect to optional stopping; this will be all important to enlarge the scope of its definition to local martingales.
\begin{equation}{\label{revp425}}\tag{3}\mbox{}\end{equation}
Proposition \ref{revp425}. (Revuz and Yor \cite{rev}). If \(T\) is a stopping time, the we have \(X\bullet M^{T}=(X\cdot 1_{[0,T]})\bullet M=(X\bullet M)^{T}\).
Proof. For \(N\in {\cal H}_{c}^{2}\), we have
\[\langle M^{T},N\rangle =\langle M,N\rangle^{T}=1_{[0,T]}\bullet\langle M,N\rangle =\langle 1_{[0,T]}\bullet M,N\rangle .\]
It shows that \(M^{T}=1_{[0,T]}\bullet M\). By Proposition~\ref{revp424}, we have on the one hand
\[X\bullet M^{T}=X\bullet (1_{[0,T]}\bullet M)=(X\cdot 1_{[0,T]})\bullet M,\]
and on the other hand
\[(X\bullet M)^{T}=1_{[0,T]}\bullet (X\bullet M)=(1_{[0,T]}\cdot X)\bullet M\]
which completes the proof. \(\blacksquare\)
Definition. If \(M\) is a continuous local martingale, we call \({\cal L}_{loc}^{2}(M)\) the space of classes of progressively measurable processes \(X\) for which there exists a sequence \(\{T_{n}\}\) of stopping times increasing to infinity and such that
\[E\left [\int_{0}^{T_{n}}X_{s}^{2}d\langle M\rangle_{s}\right ]<\infty . \sharp\]
Observe that \({\cal L}_{loc}^{2}(M)\) consists of all the progressively measurable processes \(X\) such that
\[\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}<\infty\mbox{ for every }t.\]
\begin{equation}{\label{revp427}}\tag{}\mbox{4}\end{equation}
Theorem \ref{revp427}. (Revuz and Yor \cite{rev}). For any \(X\in {\cal L}_{loc}^{2}(M)\), there exists a unique continuous local martingale vanishing at \(0\) denoted by \(X\bullet M\) such that for any
continuous local martingale \(N\), \(\langle X\bullet M,N\rangle =X\bullet\langle M,N\rangle\).
Proof. One can choose stopping times \(T_{n}\) increasing to infinity and such that \(M^{T_{n}}\) is in \({\cal H}_{c}^{2}\) and \(X^{T_{n}}\in {\cal L}^{2}(M^{T_{n}})\). Thus, for each \(n\), we can define the stochastic integral \(Y^{(n)}=X^{T_{n}}\bullet M^{T_{n}}\). But, by Proposition~\ref{revp425}, \(Y^{(n+1)}\) coincides with \(Y^{(n)}\) on \([0,T_{n}]\); therefore, one can define unambiguously a process \(X\bullet M\) by stipulating that it is equal to \(Y^{(n)}\) on \([0,T_{n}]\). This process is obviously a continuous local martingale and, by localization, it is easily seen that \(\langle X\bullet M,N\rangle = X\bullet\langle M,N\rangle\) for every local martingale \(N\). \(\blacksquare\)
To prove that a continuous local martingale \(L\) is equal to \(X\bullet M\), it is enough to check the equality \(\langle L,N\rangle =X\bullet\langle M,N\rangle\) for all bounded \(N\)’s. Again \(X\bullet M\) is called the {\bf stochastic integral} of \(X\) with respect to \(M\) and is alternatively written
\[(X\bullet M)_{t}=\int_{0}^{t}X_{s}dM_{s}.\]
Plainly, Propositions \ref{revp424} and \ref{revp425} carry over to the general case after the obvious changes.
Stochastic Integrals for Semimartingales.
Definition. A progressively measurable process \(K\) is {\bf locally bounded} if there exists a sequence \(\{T_{n}\}_{n\in {\bf N}}\) of stopping times increasing to infinity and constants \(c_{n}\) such that \(|K^{T_{n}}|\leq c_{n}\). \(\sharp\)
All continuous adapted processes \(X\) are seen to be locally bounded by taking \(T_{n}=\inf\{t:|X_{t}|\geq n\}\). Locally bounded processes are in \({\cal L}_{loc}^{2}(M)\) for every continuous local martingale \(M\).
Definition. If \(K\) is locally bounded and \(X=M+A\) is a continuous semimartingale, the stochastic integral of \(K\) with respect to \(X\) is the continuous semimartingale
\[K\bullet X=K\bullet M+K\bullet A\]
where \(K\bullet M\) is the integral of Theorem~\ref{revp427} and \(K\bullet A\) is the pathwise Stieltjes integral with respect to \(dA\). The semimartingale \(K\bullet X\) is also written as
\[(K\bullet X)_{t}=\int_{0}^{t}K_{s}dX_{s}. \sharp\]
Proposition. The map \(K\mapsto K\bullet X\) enjoys the following properties
(i) \(H\bullet (K\bullet X)=(HK)\bullet X\) for any pair \(H,K\) of locally bounded processes;
(ii) \((K\bullet X)^{T}=(K\cdot 1_{[0,T]})\bullet X=K\bullet X^{T}\) for every stopping time \(T\);
(iii) if \(X\) is a local martingale or a process of finite variation, so is \(K\bullet X\);
(iv) if \(K\in {\cal S}\), then if \(t_{n}\leq t<t_{n+1}\),
\[(K\bullet X)_{t}=\sum_{i=0}^{n}K_{i}(X_{t_{i+1}}-X_{t_{i}})+K_{n}(X_{t}-X_{t_{n}}). \sharp\]
If \(X\) is a \(\mathbb{P}\)-semimartingale and \(\mathbb{Q}\ll\mathbb{P}\), then \(X\) is also a \(\mathbb{Q}\)-semimartingale. Since a sequence converging in probability for \(\mathbb{P}\)
converges also in probability for \(\mathbb{Q}\), the stochastic integral for \(\mathbb{P}\) of a bounded process is \(\mathbb{Q}\)-indistinguishable from its stochastic integral for \(\mathbb{Q}\). Likewise, if we replace the filtration by another one for which \(X\) is still a semimartingle, the stochastic integrals of processes which are progressively measurable for both filtrations are the same. We now turn to a very important property of stochastic integrals, namely the counterpart of the Lebesgue dominated convergence theorem.
\begin{equation}{\label{revt4212}}\tag{5}\mbox{}\end{equation}
Theorem \ref{revt4212}. (Revuz and Yor \cite{rev}). Let \(X\) be a continuous semimartingale. If \(\{K^{(n)}\}_{n\in {\bf N}}\) is a sequence of locally bounded processes converging to zero pointwise and if
there exists a locally bounded process \(K\) such that \(|K^{(n)}|\leq K\) for every \(n\), then \(K^{(n)}\bullet X\) converges to zero in probability, uniformly on every compact interval. \(\sharp\)
\begin{equation}{\label{revp4213}}\tag{6}\mbox{}\end{equation}
Proposition \ref{revp4213}. (Revuz and Yor \cite{rev}). If \(K\) is left-continuous and \(\{\pi_{n}\}_{n\in {\bf N}}\) is a sequence of partitions of \([0,t]\) such that \(\parallel\pi_{n}\parallel\rightarrow 0\), then
\[\int_{0}^{t}K_{s}dX_{s}=P\mbox{-}\lim_{n\rightarrow\infty}\sum_{t_{i}\in\pi_{n}}K_{t_{i}}(X_{t_{i+1}}-X_{t_{i}}).\]
Proof. If \(K\) is bounded, the right-hand side sums are the stochastic integrals of the simple processes \(\sum K_{t_{i}}\cdot 1_{(t_{i},t_{i+1}]}\) which converge pointwise to \(K\) and are bounded by \(\parallel K\parallel_{\infty}\); therefore, the result follows from Theorem~\ref{revt4212}. The general case is obtained by the use of localization. \(\blacksquare\)
Ito’s Formula.
Proposition. (Integration by Parts). If \(X\) and \(Y\) are two continuous semimartingales, then
\[X_{t}Y_{t}=X_{0}Y_{0}+\int_{0}^{t}X_{s}dY_{s}+\int_{0}^{t}Y_{s}dX_{s}+\langle X,Y\rangle_{t};\]
In particular,
\[X_{t}^{2}=X_{0}^{2}+2\int_{0}^{t}X_{s}dX_{s}+\langle X\rangle_{t}.\]
Proof. It is enough to prove the particular case which implies the general case by polarization. If \(\pi\) is a partition of \([0,t]\), we have
\[\sum_{i}(X_{t_{i+1}}-X_{t_{i}})^{2}=X_{t}^{2}-X_{0}^{2}-2\sum_{i}X_{t_{i}}(X_{t_{i+1}}-X_{t_{i}});\]
letting \(\parallel\pi\parallel\) tend to zero and using, one one hand the definition of \(\langle X\rangle\), on the other hand Proposition \ref{revp4213}, we get the desired result. \(\blakcsquare\)
If \(X\) and \(Y\) are of finite variation, this formula boils down to the ordinary integration by parts formula for Steiltjec integrals. The same will be true for the following change of variables formula. Let us also
observe that if \(M\) is a local martingale, we have
\begin{equation}{\label{reveq*99}}\tag{7}
M_{t}^{2}-\langle M\rangle_{t}=M_{0}^{2}+2\int_{0}^{t}M_{s}dM_{s};
\end{equation}
we already knew that \(M^{2}-\langle M\rangle\) is a local martingale but the above formula gives us an explicit expression of this local martingale.
A \(n\)-dimensional vector local martingale (resp. vector continuous semimartingale) is a \(\mathbb{R}^{n}\)-valued process \({\bf X}=(X^{(1)},\cdots , X^{(n)})\) such that \(X^{i}\) is a local martingale (resp. continuous semimartingale). A complex local martingale (resp. complex continuous semimartingale) is a \({\bf C}\)-valued process whose real and imaginary parts are local martingales (resp. continuous semimartingales).
\begin{equation}{\label{revt433}}\tag{8}\mbox{}\end{equation}
Theorem. \ref{revt433}. (Ito’s Formula). Let \({\bf X}=(X^{(1)},\cdots ,X^{(n)})\) be a continuous vector semimartingale and \(f\in C^{2}(\mathbb{R}^{n})\); then, \(f({\bf X})\) is a continuous semimartingale and
\[f({\bf X}_{t})=f({\bf X}_{0})+\sum_{i}\int_{0}^{t}\frac{\partial f}{\partial x_{i}}({\bf X}_{s})dX_{s}^{(i)}+\frac{1}{2}\sum_{i,j}\int_{0}^{t}
\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf X}_{s})d\langle X^{(i)},X^{(j)}\rangle_{s}. \sharp\]
We give some remarks as follows.
- (a) The differentiability properties of \(f\) may be somewhat relaxed. For instance, if some of the \(X^{(i)}\)’s are of finite variation, \(f\) needs only be of class \(C^{1}\) in the corresponding coordinates; the proof goes through just the same. In particular, if \(X\) is a continuous semimartingales and \(A\in {\cal A}\), and if \(\partial^{2}f/\partial x^{2}\) and \(\partial f/\partial y\) exist and continuous, then
\[f(X_{t},A_{t})=f(X_{0},A_{0})+\int_{0}^{t}\frac{\partial f}{\partial x}(X_{s},A_{s})dX_{s}+\int_{0}^{t}\frac{\partial f}{\partial y}(X_{s},A_{s})
dA_{s}+\frac{1}{2}\int_{0}^{t}\frac{\partial^{2}f}{\partial x^{2}}(X_{s},A_{s})d\langle X\rangle_{s}.\] - (b) Ito’s formula may be written in “differential” form
\[df({\bf X}_{t})=\sum_{i}\frac{\partial f}{\partial x_{i}}({\bf X}_{t})dX_{t}^{(i)}+\frac{1}{2}\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}({\bf X}_{t})d\langle X^{(i)},X^{(j)}\rangle_{t}.\]
More generally, if \({\bf X}\) is a vector semimartingale, \(dY_{t}=\sum_{i}H_{t}^{(i)}dX_{t}^{(i)}\) will mean
\[Y_{t}=Y_{0}+\sum_{i}\int_{0}^{t}H_{s}^{(i)}dX_{s}^{(i)}.\]
In this setting, Ito’s formula may be read as “the chain rule for stochastic differentials”. - (c) Ito’s formula shows precisely that the class of semimartingales is invariant under composition with \(C^{2}\)-functions, which gives another reason for the introduction of semimartingales. If \(M\) is a local martingale, or even a martingale, \(f(M)\) is usually not a local martinmgale but only a semimartingale.
- (d) Let \(\phi\) be a \(C^{1}\)-function with compact support in \((0,1)\). It is of finite variation, hence may be looked upon as a semimartingale and the intgration by parts formula yields
\[X_{1}\phi (1)=X_{0}\phi (0)+\int_{0}^{1}\phi (s)dX_{s}+\int_{0}^{1}X_{s}\phi^{\prime}(s)ds+\langle X,\phi\rangle_{1}\]
which reduces to
\[\int_{0}^{1}\phi (s)dX_{s}=-\int_{0}^{1}X_{s}\phi^{\prime}(s)ds.\]
In the following result, we introduce the class of {\bf exponential local martingales} which turns out to be very important.
Proposition. If \(f\) is a complex-valued function defined on \(\mathbb{R}\times \mathbb{R}_{+}\), and such that \(\partial^{2}f/\partial x^{2}\) and \(\partial f/\partial y\) exist, are continuous and satisfy
\[\frac{\partial f}{\partial y}+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}=0,\]
then for any continuous local martingale \(M\), the process \(f(M_{t},\langle M\rangle_{t})\) is a local martingale. In particular for any \(\lambda\in {\bf C}\), the process
\[{\cal E}^{\lambda}(M)_{t}=\exp\left (\lambda M_{t}-\frac{\lambda^{2}}{2}\langle M\rangle_{t}\right )\]
is a local martingale. For \(\lambda =1\), we write simply \({\cal E}(M)\) and speak of the exponential of \(M\).
Proof. This follows at once by making \(A=\langle M\rangle\) in remark (a) below Thoerem \ref{revt433}.
We give some remarks as follows.
- (a) For Brownian motion \(W\), we already knew that \(\exp\left (\lambda W_{t}- \frac{\lambda^{2}}{2}t\right )\) is a martingale. Let us further observe that, for \(f\in {\cal L}_{loc}^{2}(\mathbb{R}_{+})\), the exponential
\[{\cal E}_{t}^{f}=\exp\left (\int_{0}^{t}f(s)dW_{s}-\frac{1}{2}\int_{0}^{t}f^{2}(s)ds\right )\] is a martingale; this follows easily from the fact that \(\int_{s}^{t} f(u)dW_{u}\) is a centered Gaussian random variable with variance \(\int_{s}^{t}f^{2}(u)du\) and is independent of \({\cal F}_{s}\). Likewise for \(n\)-dimensional Brownian motion and a \(n\)-tuple \({\bf f}=(f_{1},\cdots , f_{n})\) of functions in \({\cal L}_{loc}^{2}(\mathbb{R}_{+})\), and for the same reason,
\[{\cal E}_{t}^{{\bf f}}=\exp\left (\sum_{k=1}^{n}\int_{0}^{t}f_{k}(s)dW_{s}^{(k)}-\frac{1}{2}\sum_{k=1}^{n}\int_{0}^{t}f_{k}^{2}(s)ds\right )\]
is a martingale. - (b) If \(M_{0}=0\) then \({\cal E}^{\lambda}(M)\) is a martingale if and only if \(E[{\cal E}^{\lambda}(M)_{t}]=1\). The necessity is clear and the sufficiency comes from the fact that a supermartingale with constant expectation is a martingale.
- (c) For a continuous semimartingale \(X\), we can equally define
\[{\cal E}^{\lambda}(X)_{t}=\exp\left (\lambda X_{t}-\frac{\lambda^{2}}{2}\langle X\rangle_{t}\right )\]
and we still have
\[{\cal E}^{\lambda}(X)_{t}={\cal E}^{\lambda}(X)_{0}+\int_{0}^{t}\lambda {\cal E}^{\lambda}(X)_{s}dX_{s}.\]
This can be stated as: \({\cal E}^{\lambda}(X)\) is a solution to the stochastic differential equation
\[dY_{t}=\lambda Y_{t}dX_{t}.\]
When \(X_{0}=0\), it is in fact the unique solution such that \(Y_{0}=1\).
Proposition. If \({\bf W}\) is a \(n\)-dimensional Brownian motion and \(f\in C^{2} (\mathbb{R}_{+}\times\mathbb{R}^{n})\), then
\[M_{t}^{f}=f(t,{\bf W}_{t})-\int_{0}^{t}\left (\frac{1}{2}\Delta f+\frac{\partial f}{\partial t}\right )(s,{\bf W}_{s})ds\] is a local martingale. In particular if \(f\) is harmonic in \(\mathbb{R}^{n}\) then \(f({\bf W})\) is a local martingale.
Proof. Because of their independence, the components \(W^{(i)}\) of \({\bf W}\) satisfy \(\langle W^{(i)},W^{(j)}\rangle_{t}=\delta_{ij}t\). Thus, the claim is straightforward consequence of Ito’s formula. \(\blacksquare\)
Theorem. (L\'{e}vy’s Characterization Theorem). For a \(\{{\cal F}_{t}\}\)-adapted continuous \(n\)-dimensional process \({\bf X}\) vanishing at \(0\), the following three conditions are equivalent
(a) \({\bf X}\) is an \({\cal F}_{t}\)-Brownian motion;
(b) \({\bf X}\) is a continuous martingale and \(\langle X^{(i)},X^{(j)}\rangle_{t}=\delta_{ij}t\) for every \(1\leq i,j\leq n\).
(c) \({\bf X}\) is a continuous local martingale and for every \(n\)-tuple \({\bf f}=(f_{1},\cdots ,f_{n})\) of functions in \({\cal L}^{2}(\mathbb{R}_{+})\), the process
\[{\cal E}_{t}^{i{\bf f}}=\exp\left (i\sum_{k}\int_{0}^{t}f_{k}(s)dX_{s}^{(k)}+\frac{1}{2}\sum_{k}\int_{0}^{t}f_{k}^{2}(s)ds\right )\]
is a complex martingale. \(\sharp\)
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Approach II.
This follows from Karatzas and Shreve \cite{kar}.
Stochastic Integrals for Martingales.
We now impose a metric structure on \({\cal M}_{2}\) and discuss the nature of both \({\cal M}_{2}\) and its subspace \({\cal M}_{2}^{c}\) under this metric.
Definition. For any \(X\in {\cal M}_{2}\) and \(t\geq 0\), we define
\[\parallel X\parallel_{t}\equiv\sqrt{E[X_{t}^{2}]}.\]
We also set
\[\parallel X\parallel\equiv\sum_{n=1}^{\infty}\frac{\parallel X\parallel_{n}\wedge 1}{2^{n}}. \sharp\]
Let us observe that the function \(t\mapsto\parallel X\parallel_{t}\) on \([0,\infty )\) is nondecreasing because \(X^{2}\) is a submartingale. Further, \(\parallel X-Y\parallel\) is a pseudo-metric on \({\cal M}_{2}\), which becomes a metric if we identify indistinguishable processes. Indeed, suppose that for \(X,Y\in {\cal M}_{2}\) we have \(\parallel X-Y\parallel =0\); this implies \(X_{n}=Y_{n}\) \(P\)-a.s. for every \(n\geq 1\), and thus \(X_{t}=E[X_{n}|{\cal F}_{t}]=E[Y_{n}|{\cal F}_{t}]=Y_{t}\) \(P\)-a.s. for every \(0\leq t\leq n\). Since \(X\) and \(Y\) are right-continuous, they are indistinguishable.
\begin{equation}{\label{karp1523}}\tag{}\mbox{9}\end{equation}
Proposition \ref{karp1523}. (Karatzas and Shreve \cite{kar}). Under the preceding metric, \({\cal M}_{2}\) is a complete metric space, and \({\cal M}_{2}^{c}\) a closed subspace of \({\cal M}_{2}\). \(\sharp\)
Let us consider a continuous, square-integrable martingale \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) on a probability space \((\Omega ,{\cal F},P)\) equipped with the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\), which will be assumed to satisfy the usual conditions. We now consider what kinds of integrands are appropriate for the following integral
\begin{equation}{\label{kareq321}}\tag{10}
I_{t}(X)=\int_{0}^{t}X_{s}(\omega )dM_{s}(\omega ).
\end{equation}
We first define a measure \(\mu_{M}\) on \(([0,\infty )\times\Omega ,{\cal B}\times {\cal F})\) by setting
\[\mu_{M}(A)=E\left [\int_{0}^{\infty}1_{A}(t,\omega )d\langle M\rangle_{t}(\omega )\right ].\]
We shall say that two measurable, adapted processes \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) and \(\{(Y_{t},{\cal F}_{t})\}_{t\geq 0}\) are {\bf equivalent} if \(X_{t}(\omega )=Y_{t}(\omega )\) \(\mu_{M}\)-a.e. \((t,\omega )\). This introduces an equivalence relation. For a measurable \({\cal F}_{t}\)-adapted process \(X\), we define
\begin{equation}{\label{kareq323}}\tag{11}
[\! |X|\!]_{t}^{2}\equiv E\left [\int_{0}^{t}X_{s}^{2}(\omega )d\langle M\rangle_{s}(\omega )\right ],
\end{equation}
provided that the right-hand side is finite. Then \([\! |X|\!]_{t}\) is the \(L^{2}\)-norm for \(X\), regarded as a function of \((t,\omega )\) restricted to the space \([0,t]\times\Omega\), under the measure \(\mu_{M}\). We have \([\! |X-Y|\!]_{t}=0\) for all \(t>0\) if and only if \(X\) and \(Y\) are equivalent.
\begin{equation}{\label{kard321}}\tag{12}\mbox{}\end{equation}
Definition \ref{kard321}. Let \({\cal L}\) denote the set of equivalence classes of all measurable, \({\cal F}_{t}\)-adapted processes \(X\), for which \([\! |X|\!]_{t}<\infty\) for all \(t>0\), We define a metric on \({\cal L}\) by \([\! |X-Y|\!]\), where
\[[\! |X|\!]\equiv\sum_{n=1}^{\infty}\frac{[\! |X|\!]_{n}\wedge 1}{2^{n}}.\]
Let \({\cal L}^{*}\) denote the set of equivalence classes of progressively measurable processes satisfying \([\! |X|\!]_{t}<\infty\) for all \(t>0\), and define a metric on \({\cal L}^{*}\) in the same way. \(\sharp\)
Note that \({\cal L}\) (resp. \({\cal L}^{*}\)) contains all bounded, measurable, \({\cal F}_{t}\)-adapted (resp. bounded, progressively measurable) processes. Both \({\cal L}\) and \({\cal L}^{*}\) depend on the martingale \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\). When we wish to indicate this dependence explicitly, we write \({\cal L}(M)\) and \({\cal L}^{*}(M)\). For \(0<t<\infty\), let \({\cal L}_{t}^{*}\) denote the class of processes \(X\) in \({\cal L}^{*}\) for which \(X_{s}(\omega )=0\) for all \(s>t\). For \(t=\infty\), \({\cal L}_{t}^{*}\) is defined as the class of processes \(X\in {\cal L}^{*}\) for which $latex E\left [\int_{0}^{t}X_{s}^{2}
d\langle M\rangle_{s}\right ]<\infty$ (a condition we already have for \(t<\infty\), by virtue of membership in \({\cal L}^{*}\)). A process \(X\in {\cal L}_{t}^{*}\) can be identified with one defined only for \((s,\omega )\in [0,t]\times\Omega\), and so we can regard \({\cal L}_{t}^{*}\) as a subspace of the Hilbert space \({\cal L}^{2}([0,t])\equiv L^{2}([0,t]\times\Omega ,{\cal B}\times {\cal F}_{t},\mu_{M})\). More precisely, we regard an equivalence class in \({\cal L}^{2}([0,t])\) as a member of \({\cal L}_{t}^{*}\) if it contains a progressively measurable representative.
Proposition. For \(0<t\leq\infty\), \({\cal L}_{t}^{*}\) is a closed subspace of \({\cal L}^{2}([0,t])\). In particular, \({\cal L}_{t}^{*}\) is complete under the norm \([\! |X|\!]_{t}\) of \((\ref{kareq323})\). \(\sharp\)
Definition. A process \(X\) is called {\bf simple} if there exists a strictly increasing sequence of real numbers \(\{t_{n}\}_{n\in {\bf N}}\) with \(t_{0}=0\) and \(\lim_{n\rightarrow\infty}t_{n}=\infty\), as well as a sequence of random variables \(\{\xi_{n}\}_{n\in {\bf N}}\) with \(\sup_{n\geq 0}|\xi_{n}(\omega )|\leq c<\infty\) for every \(\omega\in\Omega\) such that \(\xi_{n}\) is \({\cal F}_{t_{n}}\)-measurable for every \(n\geq 0\) and
\[X_{t}(\omega )=\xi_{0}(\omega )\cdot 1_{\{0\}}(t)+\sum_{i=0}^{\infty}\xi_{i}(\omega )\cdot 1_{(t_{i},t_{i+1}]}(t)\mbox{ for }0\leq t,\omega\in\Omega .\]
The class of all simple processes will be denoted by \({\cal L}_{0}\). \(\sharp\)
Note that, because members of \({\cal L}_{0}\) are progressively measurable and bounded, we have \({\cal L}_{0}\subset {\cal L}^{*}\subset {\cal L}\). The construction of the stochastic integral (\ref{kareq321}) can now be outlined as follows. The integral is defined in the obvious way for \(X\in {\cal L}_{0}\) as a martingale transform
\[I_{t}(X)\equiv\sum_{i=1}^{n-1}\xi_{i}\cdot (M_{t_{i+1}}-M_{t_{i}})+\xi_{n}\cdot (M_{t}-M_{t_{n}})=\sum_{i=0}^{\infty}\xi_{i}\cdot (M_{t\wedge t_{i+1}}-M_{t\wedge t_{i}})\mbox{ for }t\geq 0,\]
where \(n\geq 0\) is the unique integer for which \(t_{n}\leq t<t_{n+1}\). The definition is then extended to integrands \(X\in {\cal L}^{*}\) and \(X\in {\cal L}\), thanks to the crucial results which show that elements of \({\cal L}^{*}\) and \({\cal L}\) can be approximated, in a suitable sense, by simple processes.
Lemma. Let \(X\) be a bounded, measurable, \({\cal F}_{t}\)-adapted process. Then there exists a sequence \(\{X^{(n)}\}_{n\in {\bf N}}\) of simple processes satisfying
\[\sup_{t>0}\lim_{n\rightarrow\infty} E\left [\int_{0}^{t}\left |X^{(n)}_{s}-X_{s}\right |^{2}ds\right ]=0. \sharp\]
\begin{equation}{\label{karp326}}\tag{13}\mbox{}\end{equation}
Proposition \ref{karp326}. (Karatzas and Shreve \cite{kar}). If the function \(t\mapsto\langle M\rangle_{t}(\omega )\) is absolutely continuous with respect to Lebesgue measure for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), then \({\cal L}_{0}\) is dense in \({\cal L}\) with respect to the metric of Definition \ref{kard321}. \(\sharp\)
When \(t\mapsto\langle M\rangle_{t}\) is not an absolutely continuous function of the time variable \(t\), we simply choose a more convenient clock.
Lemma. Let \(\{A_{t}\}_{t\geq 0}\) be a continuous, increasing process adapted to the filtration of the martingale \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\). If \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) is a progressively measurable process satisfying
\[\mathbb{E}\left [\int_{0}^{t}X_{s}^{2}dA_{s}\right ]<\infty\]
for each \(t>0\), then there exists a sequence \(\{X^{(n)}\}_{n\in {\bf N}}\) of simple processes such that
\[\sup_{t>0}\lim_{n\rightarrow\infty}E\left [\int_{0}^{t}\left |X_{s}^{(n)}-X_{s}\right |^{2}dA_{s}\right ]=0. \sharp\]
\begin{equation}{\label{karp328}}\tag{14}\mbox{}\end{equation}
Proposition \ref{karp328}. (Karatzas and Shreve \cite{kar}). The set \({\cal L}_{0}\) of simple processes is dense in \({\cal L}^{*}\) with respect to the metric of Definition~\ref{kard321}. \(\sharp\)
Proposition. For \(X,Y\in {\cal L}_{0}\) and \(0\leq s<t<\infty\), we have
(i) \(I_{0}(X)=0\) \(\mathbb{P}\)-a.s.;
(ii) \(\mathbb{E}[I_{t}(X)|{\cal F}_{s}]=I_{s}(X)\) \(P\)-a.s.;
(iii) \({\displaystyle \mathbb{E}[I_{t}^{2}(X)]=\mathbb{E}\left [\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}\right ]}\);
(iv) \(\parallel I(X)\parallel =[\! |X|\!]\);
(v) \({\displaystyle \mathbb{E}\left .\left [(I_{t}(X)-I_{s}(X))^{2}\right |{\cal F}_{s}\right ]=\mathbb{E}\left .\left [\int_{s}^{t}X_{u}^{2}d\langle M\rangle_{u}\right |{\cal F}_{s}\right ]}\) \(\mathbb{P}\)-a.s.;
(vi) \(I(\alpha X+\beta Y)=\alpha I(X)+\beta I(Y)\) for \(\alpha,\beta\in \mathbb{R}\). \(\harp\)
Property (v) establishes the fact that the continuous martingale \(I(X)\) is square-integrable with quadratic variation
\begin{equation}{\label{kareq3219}}\tag{15}
\langle I(X)\rangle_{t}=\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}.
\end{equation}
For \(X\in {\cal L}^{*}\), Proposition~\ref{karp328} implies the existence of a sequence \(\{X^{(n)}\}_{n\in {\bf N}}\subset {\cal L}_{0}\) such that \([\! |X^{(n)}-X|\!]\rightarrow 0\) as \(n\rightarrow\infty\). It follows from properties (iv) and (vi) that
\[\parallel I(X^{(n)})-I(X^{(m)})\parallel =\parallel I(X^{(n)}-X^{(m)})\parallel =[\! |X^{(n)}-X^{(m)}|\!]\rightarrow 0\]
as \(n,m\rightarrow\infty\). In other words, \(\{I(X^{(n)})\}_{n\in {\bf N}}\) is a Cauchy sequence in \({\cal M}_{2}^{c}\). By Proposition~\ref{karp1523}, there exists a process \(I(X)\) in \({\cal M}_{2}^{c}\),
defined modulo indistinguishability, such that \(\parallel I(X^{(n)})-I(X)\parallel\rightarrow 0\) as \(n\rightarrow\infty\). Because it belongs to \({\cal M}_{2}^{c}\), \(I(X)\) enjoys properties (i) and (ii). For \(0\leq s<t<\infty\), \(\{I_{s}(X^{(n)})\}_{n\in {\bf N}}\) and \(\{I_{t}(X^{(n)})\}_{n\in {\bf N}}\) converges in mean-square to \(I_{s}(X)\) and \(I_{t}(X)\), respectively; so for \(A\in {\cal F}_{s}\), property (v) applies to \(\{X^{(n)}\}_{n\in {\bf N}}\) gives
\begin{align*}
\mathbb{E}\left [1_{A}\cdot (I_{t}(X)-I_{s}(X))^{2}\right ] & =\lim_{n\rightarrow\infty}\mathbb{E}\left [1_{A}\cdot\left (I_{t}(X^{(n)})-I_{s}(X^{(n)})\right )^{2}\right ]\\
& =\lim_{n\rightarrow\infty}E\left [1_{A}\cdot\int_{s}^{t}(X_{u}^{(n)})^{2}d\langle M\rangle_{u}\right ]\\
& =\mathbb{E}\left [1_{A}\cdot\int_{s}^{t}X_{u}^{2}d\langle M\rangle_{u}\right ],
\end{align*}
where the last equality follows from \(\lim_{n\rightarrow\infty}[\! |X^{(n)}-X|\!]=0\). This proves that \(I(X)\) also satisfies property (v) and, consequently, property (iii) and (iv). Because \(X\) and \(M\) are
progressively measurable, \(\int_{s}^{t}X_{u}^{2}d\langle M\rangle_{u}\) is \({\cal F}_{t}\)-measurable for fixed \(0\leq s<t<\infty\), and so property (v) gives us (\ref{kareq3219}). The validity of property (vi) for \(X,Y\in {\cal L}^{*}\) also follows from its validity for processes in \({\cal L}_{0}\), upon passage to the limit.
The process \(I(X)\) for \(X\in {\cal L}^{*}\) is well-defined; if we have two sequences \(\{X^{(n)}\}_{n\in {\bf N}}\) and \(\{Y^{(n)}\}_{n\in {\bf N}}\) in \({\cal L}_{0}\) with the property \(\lim_{n\rightarrow\infty}[\!|X^{(n)}-X|\!]=0\), \(\lim_{n\rightarrow\infty}[\!|Y^{(n)}-Y|\!]=0\), we can construct a third sequence \(\{Z^{(n)}\}_{n\in {\bf N}}\) with this property, by setting \(Z^{(2n-1)}=X^{(n)}\) and \(Z^{(2n)}=Y^{(n)}\) for \(n\in {\bf N}\). The limit \(I(X)\) of the sequence \(\{I(Z^{(n)})\}_{n\in {\bf N}}\) in \({\cal M}_{2}^{c}\) has to agree with the limits of both sequences, namely \(\{I(X^{(n)})\}_{n\in {\bf N}}\) and \(\{I(Y^{(n)})\}_{n\in {\bf N}}\).
Definition. For \(X\in {\cal L}^{*}\), the stochastic integral of \(X\) with respect to the martingale \(M\in {\cal M}_{2}^{c}\) is the unique, square-integrable martingale \(I(X)\) which satisfies \(\lim_{n\rightarrow\infty}\parallel I(X^{(n)})-I(X)\parallel =0\) for every sequence \(\{X^{(n)}\}_{n\in {\bf N}}\subset {\cal L}_{0}\) with \(\lim_{n\rightarrow\infty}[\! |X^{(n)}-X|\!]=0\). We write
\[I_{t}(X)=\int_{0}^{t}X_{s}dM_{s}\mbox{ for }t\geq 0. \sharp\]
\begin{equation}{\label{karp3210}}\tag{16}\mbox{}\end{equation}
Proposition \ref{karp3210}. (Karatzas and Shreve \cite{kar}). For \(M\in {\cal M}_{2}^{c}\) and \(X\in {\cal L}^{*}\), the stochastic integral \(I(X)\) of \(X\) with respect to \(M\) satisfies properties (i)-(v), as well as property (vi) for every \(Y\in {\cal L}^{*}\), and has quadratic variation process given by \((\ref{kareq3219})\). Furthermore, for any two stopping times \(S\leq T\) of the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\) and any number \(t>0\), we have
\[\mathbb{E}\left .\left [I_{t\wedge T}(X)\right |{\cal F}_{S}\right ]=I_{t\wedge S}(X)\mbox{ \(\mathbb{P}\)-a.s.}\]
With \(X,Y\in {\cal L}^{*}\), we have \(P\)-a.s.
\[\mathbb{E}\left .\left [(I_{t\wedge T}(X)-I_{t\wedge S}(X))\cdot (I_{t\wedge T}(Y)-I_{t\wedge S}(Y))\right |{\cal F}_{S}\right ]=\mathbb{E}\left .\left [\int_{t\wedge S}^{t\wedge T}X_{u}Y_{u}d\langle M\rangle_{u}\right |{\cal F}_{S}\right ],\]
and in particular, for any number \(s\in [0,t]\),
\[\mathbb{E}\left .\left [(I_{t}(X)-I_{s}(X))\cdot (I_{t}(Y)-I_{s}(Y))\right |{\cal F}_{s}\right ]=E\left .\left [\int_{s}^{t}X_{u}Y_{u}d\langle M\rangle_{u}\right |{\cal F}_{s}\right ].\]
Finally
\[I_{t\wedge T}(X)=I_{t}(\widetilde{X})\mbox{ \(\mathbb{P}\)-a.s.,}\]
where \(\widetilde{X}_{t}(\omega )\equiv X_{t}(\omega )\cdot 1_{\{t\leq T(\omega )\}}\). \(\sharp\)
If the sample paths \(t\mapsto\langle M\rangle_{t}(\omega )\) of the quadratic variation process \(\langle M\rangle\) are absolutely continuous functions of \(t\) for \(\mathbb{P}\)-a.s., the Proposition \ref{karp326} can be used instead of Proposition \ref{karp328} to define \(I(X)\) for every \(X\in {\cal L}\). We have \(I(X)\in {\cal M}_{2}^{c}\) and all the properties of Proposition~\ref{karp3210} in this case.
Proposition. (An Inequality of Kunita and Watanabe). If \(M,N\in {\cal M}_{2}^{c}\), \(X\in {\cal L}^{*}(M)\) and \(Y\in {\cal L}^{*}(N)\), then a.s.
\[\int_{0}^{t}|X_{s}Y_{s}|d\xi_{s}\leq\left (\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}\right )^{1/2}\cdot\left (\int_{0}^{t}Y_{s}^{2}d\langle N\rangle_{s}\right )^{1/2}\mbox{ for }t\geq 0,\]
where \(\xi_{s}\) denotes the total variation of the process \(\langle M,N\rangle\) on \([0,s]\). \(\sharp\)
Suppose \(M\) and \(N\) are in \({\cal M}_{2}^{c}\) and take \(X\in {\cal L}^{*}(M),Y\in {\cal L}^{*}(N)\). Then
\[I_{t}^{M}(X)\equiv\int_{0}^{t}X_{s}dM_{s}\mbox{ and }I_{t}^{N}(Y)\equiv\int_{0}^{t}Y_{s}dN_{s}\]
are also in \({\cal M}_{2}^{c}\) and, according to (\ref{kareq3219}),
\[\langle I^{M}(X)\rangle_{t}=\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}\mbox{ and }\langle I^{N}(Y)\rangle_{t}=\int_{0}^{t}Y_{s}^{2}d\langle N\rangle_{s}.\]
We propose now to establish the cross-variation formula
\begin{equation}{\label{kareq3226}}\tag{17}
\langle I^{M}(X),I^{N}(Y)\rangle_{t}=\int_{0}^{t}X_{s}Y_{s}d\langle M,N\rangle_{s}\mbox{ for }t\geq 0\mbox{ \(P\)-a.s.}
\end{equation}
If \(X\) and \(Y\) are simple, then it is straightforward to show by computation that for \(0\leq s<t<\infty\),
\begin{equation}{\label{kareq3227}}\tag{18}
\mathbb{E}\left .\left [\left (I_{t}^{M}(X)-I_{s}^{M}(X)\right )\cdot\left (I_{t}^{N}(Y)-I_{s}^{N}(Y)\right )\right |{\cal F}_{s}\right ]=E\left .\left [\int_{s}^{t}X_{u}Y_{u}d\langle M,N\rangle_{u}\right |{\cal F}_{s}\right ]\mbox{ \(\mathbb{P}\)-a.s.}
\end{equation}
This is equivalent to (\ref{kareq3226}). It remains to extend this result from simple processes to the case of \(X\in {\cal L}^{*}(M)\) and \(Y\in {\cal L}^{*}(N)\).
Lemma. If \(M,N\in {\cal M}_{2}^{c}\), \(X\in {\cal L}^{*}(M)\), and \(\{X^{(n)}\}_{n\in {\bf N}}\subset {\cal L}^{*}(M)\) is such that for some \(t>0\)
\[\lim_{n\rightarrow\infty}\int_{0}^{t}\left |X_{s}^{(n)}-X_{s}\right |^{2}d\langle M\rangle_{s}=0\mbox{ \(\mathbb{P}\)-a.s.,}\]
then
\[\lim_{n\rightarrow\infty}\langle I(X^{(n)}),N\rangle_{s}=\langle I(X),N\rangle_{s}\mbox{ for \(0\leq s\leq t\) \(\mathbb{P}\)-a.s.} \sharp\]
Lemma. If \(M,N\in {\cal M}_{2}^{c}\) and \(X\in {\cal L}^{*}(M)\), then
\[\langle I^{M}(X),N\rangle_{t}=\int_{0}^{t}X_{s}d\langle M,N\rangle_{s}\mbox{ for \(t\geq 0\) \(\mathbb{P}\)-a.s.} \ㄐㄣ\]
Proposition. If \(M,N\in {\cal M}_{2}^{c}\), \(X\in {\cal L}^{*}(M)\) and \(Y\in {\cal L}^{*}(N)\), then the equivalent formulas (\ref{kareq3226}) and (\ref{kareq3227}) hold. \(\sharp\)
Proposition. Consider a martingale \(M\in {\cal M}_{2}^{c}\) and a process \(X\in {\cal L}^{*}(M)\). The stochastic integral \(I^{M}(X)\) is the unique martingale \(Z\in {\cal M}_{2}^{c}\) which satisfies
\begin{equation}{\label{kareq3230}}\tag{19}
\langle Z,N\rangle_{t}=\int_{0}^{t}X_{s}d\langle M,N\rangle_{s}\mbox{ for \(t\geq 0\) \(\mathbb{P}\)-a.s.,}
\end{equation}
for every \(N\in {\cal M}_{2}^{c}\). \(\sharp\)
\begin{equation}{\label{karc3220}}\tag{20}\mbox{}\end{equation}
Corollary \ref{karc3220}. (Karatzas and Shreve \cite{kar}). Suppose \(M\in {\cal M}_{2}^{c}\), \(X\in {\cal L}^{*}(M)\) and \(N=I^{M}(X)\). Suppose further that \(Y\in {\cal L}^{*}(N)\). Then \(XY\in {\cal L}^{*}(M)\) and \(I^{N}(Y)=I^{M}(XY)\). \(\sharp\)
\begin{equation}{\label{karc3221}}\tag{21}\mbox{}\end{equation}
Corollary \ref{karc3221}. (Karatzas and Shreve \cite{kar}). Suppose \(M,\widetilde{M}\in {\cal M}_{2}^{c}\), \(X\in {\cal L}^{*}(M)\) and \(\widetilde{X}\in {\cal L}^{*}(\widetilde{M})\), and there exists a stopping time \(T\) of the common filtration for these processes such that for \(P\)-a.e. \(\omega\), \(X_{t\wedge T(\omega )}(\omega )=\widetilde{X}_{t\wedge T(\omega )}(\omega )\) and \(M_{t\wedge T(\omega )}(\omega )=\widetilde{M}_{t\wedge T(\omega )}(\omega )\) for \(t\geq 0\). Then \(I_{t\wedge T(\omega )}^{M}(X)(\omega )= I_{t\wedge T(\omega )}^{\widetilde{M}}(\widetilde{X})(\omega )\) for \(t\geq 0\) \(\mathbb{P}\)-a.e. \(\omega\). \(\sharp\)
Stochastic Integrals for Local Martingales.
Corollary \ref{karc3221} shows that stochastic integrals are determined locally by the local values of the integrator and integrand. This fact allows us to broaden the classes of both integrators and integrands. Let \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) be a continuous local martingale on a probability space \((\Omega ,{\cal F},P)\) with \(M_{0}=0\) a.s. Recall the assumption that \(\{{\cal F}_{t}\}_{t\geq 0}\) satisfies the usual conditions.
Definition. We denote by \({\cal P}\) the collection of equivalence classes of all measurable, adapted processes \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) satisfying
\begin{equation}{\label{kareq3231}}\tag{22}
\mathbb{P}\left\{\int_{0}^{t}X_{s}^{2}d\langle M\rangle_{s}<\infty\right\}=1\mbox{ for every }t\geq 0.
\end{equation}
We denote by \({\cal P}^{*}\) the collection of equivalence classes of all progressively measurable processes satisfying this condition. \(\sharp\)
We shall continue the development only for integrands in \({\cal P}^{*}\). If a.e. path \(t\mapsto\langle M\rangle_{t}(\omega )\) of the quadratic variation process \(\langle M\rangle\) is an absolutely continuous function, we can choose integrands from the wider class \({\cal P}\). Because \(M\) is a continuous local martingale, there is a nondecreasing sequence \(\{S_{n}\}_{n\in {\bf N}}\) of stopping times of \(\{{\cal F}_{t}\}_{t\geq 0}\) such that \(\lim_{n\rightarrow\infty}S_{n}=\infty\) \(P\)-a.s., and \(\{(M_{t\wedge S_{n}},{\cal F}_{t})\}_{t\geq 0}\) is in \({\cal M}_{2}^{c}\). For \(X\in {\cal P}^{*}\), one constructs another sequence of bounded stopping times by setting
\[R_{n}(\omega )=n\wedge\inf\left\{t\geq 0:\int_{0}^{t}X_{s}^{2}(\omega )d\langle M\rangle_{s}(\omega )\geq n\right\}.\]
This is also a nondecreasing sequence and, because of (\ref{kareq3231}), \(\lim_{n\rightarrow\infty}R_{n}=\infty\) \(P\)-a.s. For \(n\in {\bf N}\), set \(T_{n}(\omega )=R_{n}(\omega )\wedge S_{n}(\omega )\),
$M_{t}^{(n)}(\omega )=M_{t\wedge T_{n}}(\omega )$, \(X_{t}^{(n)}(\omega )=X_{t}(\omega )\cdot 1_{\{T_{n}(\omega )\geq t\}}\) for \(t\geq 0\). Then \(M^{(n)}\in {\cal M}_{2}^{c}\) and $latex X^{(n)}\in
{\cal L}^{*}(M^{(n)})$ for \(n\in {\bf N}\), so \(I^{M^{(n)}}(X^{(n)})\) is defined. Corollary~\ref{karc3221} implies that for \(1\leq n\leq m\)
\[I_{t}^{M^{(n)}}(X^{(n)})=I_{t}^{M^{(m)}}(X^{(m)})\mbox{ for }0\leq t\leq T_{n}\]
since \(\{T_{n}\}_{n\in {\bf N}}\) is a nondecreasing sequence, so we may define the stochastic integral as
\begin{equation}{\label{kareq3234}}\tag{23}
I_{t}(X)=I_{t}^{M^{(n)}}(X^{(n)})\mbox{ on }\{0\leq t\leq T_{n}\}.
\end{equation}
This definition is consistent, is independent of the choice of \(\{S_{n}\}_{n\in {\bf N}}\), and determines a continuous process, which is also a local martingale.
Definition. Let \(M\) be a continuous local martingale and \(X\in {\cal P}^{*}\), the stochastic integral of \(X\) with respect to \(M\) is the process \(\{(I_{t}(X),{\cal F}_{t})\}_{t\geq 0}\) which is a continuous local martingale defined by \((\ref{kareq3234})\). As before, we often write \(\int_{0}^{t}X_{s}dM_{s}\) instead of \(I_{t}(X)\). \(\sharp\)
When \(M\) is a continuous local martingale and \(X\in {\cal P}^{*}\), the integral \(I(X)\) will not in general satisfy properties (ii)-(v). However, the sample path properties (i), (vi) and (\ref{kareq3219}) are still valid can be easily proved by localization.
Proposition. Consider a continuous local martingale \(M\) and a process \(X\in {\cal P}^{*}(M)\). The stochastic integral \(I^{M}(X)\) is the unique continuous local martingale \(Z\) which satisfies \((\ref{kareq3230})\) for every \(N\in {\cal M}_{2}^{c}\) \((\)or equivalently, for every continuous local martingale \(N)\). \(\sharp\)
Proposition. Suppose that \(M\) and \(N\) are continuous local martingales and \(X\in {\cal P}^{*}(M)\cap {\cal P}^{*}(N)\). Then we have
\[I^{\alpha M+\beta N}(X)=\alpha\cdot I^{M}(X)+\beta\cdot I^{N}(X),\]
where \(\alpha ,\beta\) are real numbers. \(\sharp\)
Proposition. Let \(M\) be a continuous local martingale and choose \(X\in {\cal P}^{*}\). Then there exists a sequence of simple processes \(\{X^{(n)}\}_{n\in {\bf N}}\) such that, for every \(t>0\),
\[\lim_{n\rightarrow\infty}\int_{0}^{t}\left |X_{t}^{(n)}-X_{t}\right |^{2}d\langle M\rangle_{t}=0\mbox{ and }\lim_{n\rightarrow\infty}\sup_{0\leq s\leq t}\left |I_{s}(X^{(n)})-I_{s}(X)\right |=0\]
hold \(P\)-a.s. If \(M\) is a standard one-dimensional Brownian motion, then the preceding also hold with \(X\in {\cal P}\). \(\sharp\)
Ito’s Formula.
Theorem. (Ito’s Formula). Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a function of class \(C^{2}\) and let \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) be a continuous semimartingale with decomposition \((\ref{kareq331})\). Then \(\mathbb{P}\)-a.s.
\begin{equation}{\label{kareq333}}\tag{24}
f(X_{t})=f(X_{0})+\int_{0}^{t}f'(X_{s})dM_{s}+\int_{0}^{t}f'(X_{s})dB_{s}+\frac{1}{2}\int_{0}^{t}f”(X_{s})d\langle M\rangle_{s}\mbox{ for }t\geq 0. \sharp
\end{equation}
For fixed \(\omega\) and \(t>0\), the function \(X_{s}(\omega )\) is bounded for \(0\leq s\leq t\), so \(f'(X_{s}(\omega ))\) is bounded on this interval. It follows that this stochastic integral \(\int_{0}^{t}f'(X_{s})dM_{s}\) is a continuous local martingale. The other two integrals in (\ref{kareq333}) are to be understood in the Lebesgue-Stieltjes sense, and so, as functions of the upper limit of integration, are of bounded variation. Thus, \(\{(f(X_{t}),{\cal F}_{t})\}_{t\geq 0}\) is a continuous semimartingale. Equation (\ref{kareq333}) is often written in differential notation
\[df(X_{t})=f'(X_{t})dM_{t}+f'(X_{t})dB_{t}+\frac{1}{2}f”(X_{t})d\langle M\rangle_{t}=f'(X_{t})dX_{t}+\frac{1}{2}f”(X_{t})d\langle M\rangle_{t}\]
for \(t\geq 0\). This is the “chain rule” for stochastic calculus. We have the following multi-dimensional version of Ito’s formula.
Theorem. (Ito’s Formula). Let \(\{({\bf M}_{t}=(M_{t}^{(1)},\cdots ,M_{t}^{(n)}), {\cal F}_{t})\}_{t\geq 0}\) be a vector of continuous local martingale, \(\{({\bf B}_{t}=(B_{t}^{(1)},\cdots ,B_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) a vector of adapted processes of bounded variation with \({\bf B}_{0}={\bf 0}\), and set \({\bf X}_{t}={\bf X}_{0}+{\bf M}_{t}+{\bf B}_{t}\) for \(t\geq 0\), where \({\bf X}_{0}\) is an \({\cal F}_{0}\)-measurable random vector in \(\mathbb{R}^{n}\). Let \(f(t,{\bf x}):[0,\infty )\times \mathbb{R}^{n} \rightarrow \mathbb{R}\) be of class \(C^{1,2}\). Then \(\mathbb{P}\)-a.s.
\begin{align*}
f(t,{\bf X}_{t}) & =f(0,{\bf X}_{0})+\int_{0}^{t}\frac{\partial}{\partial t}f(s,{\bf X}_{s})ds+\sum_{i=1}^{n}\int_{0}^{t}\frac{\partial}{\partial x_{i}}f(s,{\bf X}_{s})dB_{s}^{(i)}+\sum_{i=1}^{n}\int_{0}^{t}\frac{\partial}{\partial x_{i}}f(s,{\bf X}_{s})dM_{s}^{(i)}\\
& +\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\int_{0}^{t}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}f(s,{\bf X}_{s})d\langle M^{(i)},M^{(j)}\rangle_{s}
\end{align*}
for \(t\geq 0\). \(\sharp\)
\begin{equation}{\label{kare339}}\tag{25}\mbox{}\end{equation}
Example \ref{kare339}. Let \(W\) be a standard Brownian motion and \(X\in {\cal P}\). We define
\[\xi_{t}(X)\equiv\int_{0}^{t}X_{u}dW_{u}-\frac{1}{2}\int_{0}^{t}X_{u}^{2}du.\]
The process \(\{(Z_{t}=\exp (\xi_{t}(X)),{\cal F}_{t})\}_{t\geq 0}\) is a supermartingale; it is a martingale if \(X\in {\cal L}_{0}\). We now check by application of Ito’s formula that this process satisfies the stochastic integral equation
\begin{equation}{\label{kareq336}}\tag{26}
Z_{t}=1+\int_{0}^{t}Z_{s}X_{s}dW_{s}\mbox{ for }t\geq 0.
\end{equation}
Indeed, \(\{(\xi_{t},{\cal F}_{t})\}_{t\geq 0}\) is a semimartingale with local martingale part \(N_{t}\equiv\int_{0}^{t}X_{s}dW_{s}\) and bounded variation part \(B_{t}\equiv -\frac{1}{2}\int_{0}^{t}X_{s}^{2}ds\). With \(f(x)=e^{x}\), we have
\begin{align*}
Z_{t}=f(\xi_{t}) & =f(\xi_{0})+\int_{0}^{t}f'(\xi_{s})dN_{s}+\int_{0}^{t}f'(\xi_{s})dB_{s}+\frac{1}{2}\int_{0}^{t}f”(\xi_{s})d\langle N\rangle_{s}\\
& =1+\int_{0}^{t}Z_{s}X_{s}dW_{s}+\int_{0}^{t}Z_{s}\cdot\left (-\frac{X_{s}^{2}}{2}\right )ds+\frac{1}{2}\int_{0}^{t}Z_{s}X_{s}^{2}ds\\
& =1+\int_{0}^{t}Z_{s}X_{s}dW_{s}.
\end{align*}
The replacement of \(dN_{s}\) by \(X_{s}dW_{s}\) in this equation is justified by Corollary \ref{karc3220}. It is usually more convenient to perform computations like this using differential notation. We write
\[d\xi_{t}=X_{t}dW_{t}-\frac{1}{2}X_{t}^{2}dt,\]
and, to reflect the fact that the martingale part of \(\xi\) has quadartic variation with differential \(X_{t}^{2}dt\), we let \((d\xi_{t})^{2}=X_{t}^{2}dt\). One may obtain this from the formal computation
\[(d\xi_{t})^{2}=\left (X_{t}dW_{t}-\frac{1}{2}X_{t}^{2}dt\right )^{2}=X_{t}^{2}(dW_{t})^{2}-X_{t}^{3}dW_{t}dt+\frac{1}{4}X_{t}^{4}(dt)^{2}=X_{t}^{2}dt,\]
using the conventional multiplication rules \(dtdt=dtdW_{t}=dW_{t}dt=0\) and \(dW_{t}dW_{t}=dt\). With this formalism, Ito’s formula can be written as
\[df(\xi_{t})=f'(\xi_{t})d\xi_{t}+\frac{1}{2}f”(\xi_{t})(d\xi_{t})^{2},\]
and with \(f(x)=e^{x}\), we obtain
\[dZ_{t}=Z_{t}X_{t}dW_{t}-\frac{1}{2}Z_{t}X_{t}^{2}dt+\frac{1}{2}Z_{t}X_{t}^{2}dt=Z_{t}X_{t}dW_{t}.\]
Taking into account the initial condition \(Z_{0}=1\), we can then recover (\ref{kareq336}). \(\sharp\)
Example. One of the motivating forces behind the Ito calculus was a desire to understand the effects of additive noise on ordinary differential equations. Suppose, for example, that we add a noise term to the linear ordinary differential equation
\[\dot{\xi}(t)=a(t)\cdot\xi (t)\]
to obtain the stochastic differential equation
\[d\xi_{t}=a(t)\cdot\xi_{t}dt+b(t)dW_{t},\]
where \(a(t)\) and \(b(t)\) are measurable nonrandom functions satisfying
\[\int_{0}^{t}|a(s)|ds+\int_{0}^{t}b^{2}(s)ds<\infty\mbox{ for }t>0,\]
and \(W\) is a Brownian motion. Applying the Ito rule to \(X_{t}^{(1)}\cdot X_{t}^{(2)}\) with
\[X_{t}^{(1)}=\exp\left (\int_{0}^{t}a(s)ds\right )\mbox{ and }X_{t}^{(2)}=\xi_{0}+\int_{0}^{t}\left (b(s)\cdot\exp\left (-\int_{0}^{s}a(u)du\right )\right )dW_{s},\]
we see that \(\xi_{t}=X_{t}^{(1)}\cdot X_{t}^{(2)}\) solves the stochastic differential equation. Note that \(\xi_{t}\) is well-defined because, for \(t>0\),
\[\int_{0}^{t}\left (b^{2}(s)\cdot\exp\left (-2\int_{0}^{s}a(u)du\right )\right )ds\leq\exp\left (2\int_{0}^{t}|a(u)|du\right )\cdot\int_{0}^{t}b^{2}(s)ds<\infty . \sharp\]
Proposition. (Integration by Parts). Suppose we have two continuous semimartingales \(X_{t}=X_{0}+M_{t}+B_{t}\) and \(Y_{t}=Y_{0}+N_{t}+C_{t}\) for \(t\geq 0\), where \(M\) and \(N\) are continuous local martingales and \(B\) and \(C\) are adapted, continuous processes of bounded variation with \(B_{0}=C_{0}=0\) a.s. Then the following integration by parts formula holds
\[\int_{0}^{t}X_{s}dY_{s}=X_{t}Y_{t}-X_{0}Y_{0}-\int_{0}^{t}Y_{s}dX_{s}-\langle M,N\rangle . \sharp\]
Theorem. (Martingale Characterization of Brownian Motion). Let \(\{{\bf X}_{t}=(X_{t}^{(1)},\cdots ,X_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a continuous adapted process in \(\mathbb{R}^{n}\) such that, for every component \(1\leq k\leq n\), the process \(M_{t}^{(k)}\equiv X_{t}^{(k)}- X_{0}^{(k)}\) for \(t\geq 0\) is a continuous local martingale relative to \(\{{\cal F}_{t}\}_{t\geq 0}\), and the cross-variations are given by \(\langle M^{(k)},M^{(j)}\rangle_{t}=\delta_{kj}\cdot t\) for \(1\leq k,j\leq n\). Then \(\{({\bf X}_{t},{\cal F}_{t})\}_{t\geq 0}\) is a \(n\)-dimensional Brownian motion. \(\sharp\)
Representations of Continuous Martingales in Terms of Brownian Motion.
Let \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) be an adapted process on some \((\Omega ,{\cal F},P)\). We may need a \(n\)-dimensional Brownian motion independent of \(X\), but because \((\Omega ,{\cal F},P)\) may not be rich enough to support this Brownian motion, we must extend the probability space to construct this. Let \((\widehat{\Omega},\widehat{{\cal F}},\widehat{P})\) be another probability space, on which we consider a \(n\)-dimensional Brownian motion \(\{({\bf W}_{t},\widehat{{\cal F}}_{t})\}_{t\geq 0}\), set \(\widetilde{\Omega }=\Omega\times\widehat{\Omega}\), \(\widetilde{{\cal G}}={\cal F}\times\widehat{{\cal F}}\), \(\widetilde{P}=P\times\widehat{P}\), and define a new filtration by \(\widetilde{{\cal G}}_{t}={\cal F}_{t}\times\widehat{{\cal F}}_{t}\). The later may not satisfy the usual conditions, so we augment it and make it right-continuous by defining
\[\widetilde{{\cal F}}_{t}=\bigcap_{s>t}\sigma (\widetilde{{\cal G}}_{s}\cup {\cal N}),\]
where \({\cal N}\) is the collection of \(\widetilde{P}\)-null sets in \(\widetilde{{\cal G}}\). We also complete \(\widetilde{{\cal G}}\) by defining \(\widetilde{{\cal F}}=\sigma (\widetilde{{\cal G}}\cup {\cal N})\). We may extend \(X\) and \({\bf W}\) to \(\widetilde{{\cal F}}_{t}\)-adapted processes on \((\widetilde{\Omega},\widetilde{{\cal F}},\widetilde{P})\) by defining for \((\omega ,\widehat{\omega})\in\widetilde{{\Omega}}\)
\[\widetilde{X}_{t}(\omega ,\widehat{\omega})=X_{t}(\omega )\mbox{ and }\widetilde{{\bf W}}_{t}(\omega ,\widehat{\omega})={\bf W}_{t}(\widehat{\omega}).\]
Then \(\{(\widetilde{{\bf W}}_{t},\widetilde{{\cal F}}_{t})\}_{t\geq 0}\) is a \(n\)-dimensional Brownian motion, independent of \(\{(\widetilde{X}_{t},\widetilde{{\cal F}}_{t})\}_{t\geq 0}\). Indeed, \(\widetilde{{\bf W}}\) is independent of the extension to \(\widetilde{\Omega}\) of any \({\cal F}\)-measurable random variable on \(\Omega\). To simplify notation, we henceforth write \(X\) and \({\bf W}\) instead of \(\widetilde{X}\) and \(\widetilde{{\bf W}}\) in the context of extensions.
Let us recall that if \(\{(W_{t},{\cal F}_{t})\}_{t\geq 0}\) is a standard Brownian motion and \(X\) is a measurable adapted process with \(\mathbb{P}\left\{\int_{0}^{t}X_{s}^{2}ds<\infty\right\}=1\) for every \(t\geq 0\), then the stochastic integral \(I_{t}(X)=\int_{0}^{t}X_{s}dW_{s}\) is a continuous local martingale with quadratic variation process \(\langle I(X)\rangle_{t}=\int_{0}^{t}X_{s}^{2}ds\), which is an absolutely continuous function of \(t\) for \(P\)-a.s. The first representation result provides the converse to this statement.
Theorem. (Continuous Local Martingales as Stochastic Integrals with Respect to Brownian Motion). Suppose $latex \{({\bf M}_{t}=(M_{t}^{(1)},\cdots ,M_{t}^{(n)}),{\cal F}_{t})
\}_{t\geq 0}$ is defined on \((\Omega ,{\cal F},P)\), where \(M^{(i)}\) are continuous local martingales for \(1\leq i\leq n\). Suppose also that for \(1\leq i,j\leq n\), the cross-variation \(\langle M^{(i)},M^{(j)}\rangle_{t}(\omega )\) is an absolutely continuous function of \(t\) for \(P\)-a.s. \(\omega\). Then there is an extension \((\widetilde{\Omega},\widetilde{{\cal F}},\widetilde{P})\) of \((\Omega ,{\cal F},P)\) on which is defined a \(n\)-dimensional Brownian motion \(\{({\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) and a matrix $latex \{(({\bf X}_{t}^{(ik)})_{i,k=1}^{n},
\widetilde{{\cal F}}_{t})\}_{t\geq 0}$ of measurable adapted processes with
\[\widetilde{P}\left\{\int_{0}^{t}(X_{s}^{(ik)})^{2}ds<\infty\right\}=1\mbox{ for }1\leq i,k\leq n, t\geq 0,\]
such that we have \(P\)-a.s., the representations
\[M_{t}^{(i)}=\sum_{k=1}^{n}\int_{0}^{t}X_{s}^{(ik)}dW_{s}^{(k)}\mbox{ and }
\langle M^{(i)},M^{(j)}\rangle_{t}=\sum_{k=1}^{n}\int_{0}^{t}X_{s}^{(ik)}\cdot X_{s}^{(jk)}ds\mbox{ for }1\leq i,j\leq n, t\geq 0. \sharp\]
\begin{equation}{\label{kart346}}\tag{27}\mbox{}\end{equation}
Theorem \ref{kart346}. (Continuous Local Martingales as Time-Changed Brownian Motion). Let \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) be a continuous local martingale satisfy \(\lim_{t\rightarrow\infty}\langle M\rangle_{t}=\infty\) \(\mathbb{P}\)-a.s. Define, for each \(0\leq s<\infty\), the stopping time \(T(s)=\inf\{t\geq 0:\langle M\rangle_{t}>s\}\). Then the time-changed process
$\{(W_{s}\equiv M_{T(s)},{\cal G}_{s}\equiv {\cal F}_{T(s)})\}_{s\geq 0}$ is a standard one-dimensional Brownian motion. In particular, the filtration \(\{{\cal G}_{s}\}_{s\geq 0}\) satisfies the usual conditions and we have, \(\mathbb{P}\)-a.s., \(M_{t}=W_{\langle M\rangle_{t}}\) for \(t\geq 0\). \(\sharp\)
Proposition. With the assumptions and the notation of Theorem~\ref{kart346}, we have the following time-change formula for stochastic integrals. If \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) is progressively measurable and satisfies \(\int_{0}^{\infty}X_{t}^{2}d\langle M\rangle_{t}<\infty\) a.s., then the process \(\{(Y_{s}\equiv X_{T(s)},{\cal G}_{s})\}_{s\geq 0}\) is adapted and satisfies, a.s.
\[\int_{0}^{\infty}Y_{s}^{2}ds<\infty ,\]
\[\int_{0}^{t}X_{s}dM_{s}=\int_{0}^{\langle M\rangle_{t}}Y_{s}dW_{s}\mbox{ for }t\geq 0,\]
\[\int_{0}^{T(s)}X_{u}dM_{u}=\int_{0}^{s}Y_{u}dW_{u}\mbox{ for }s\geq 0. \sharp\]
We now discuss the multivariate extension of Theorem \ref{kart346}.
Theorem. (Continuous Local Martingales as Time-Changed Brownian Motion). Let \(\{({\bf M}_{t}=(M_{t}^{(1)},\cdots ,M_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a continuous adapted process, where each \(M^{(i)}\) is a continuous local martingale, \(\lim_{t\rightarrow\infty}\langle M^{(i)}\rangle_{t}=\infty\) \(P\)-a.s., and \(\langle M^{(i)},M^{(j)}\rangle_{t}=0\) for \(1\leq i\neq j\leq n\), \(t\geq 0\). Define \(T_{i}(s)=\inf\{t\geq 0:\langle M^{(i)}\rangle_{t}>s\}\) for \(1\leq i\leq n\), \(s\geq 0\), so that for each \(i\) and \(s\), the random time \(T_{i}(s)\) is a stopping time for the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\). Then the processes \(W_{s}^{(i)}\equiv M_{T_{i}(s)}^{(i)}\) for \(1\leq i\leq n\), \(s\geq 0\) are independent standard one-dimensional Brownian motions. \(\sharp\)
\begin{equation}{\label{kart3415}}\tag{28}\mbox{}\end{equation}
Theorem \ref{kart3415}. (Representation of Brownian, Square-Integrable Martingales as Stochastic Integrals). Let \(\{({\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a \(n\)-dimensional Brownian motion on \((\Omega ,{\cal F},P)\), and let \(\{{\cal F}_{t}\}_{t\geq 0}\) be the augmentation under \(P\) of the filtration \(\{{\cal F}_{t}^{\bf W}\}_{t\geq 0}\) generated by \({\bf W}\). Then, for any square-integrable martingale \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) with \(M_{0}=0\) and RCLL paths a.s., there exist progressively measurable processes \(\{(Y_{t}^{(j)},{\cal F}_{t})\}_{t\geq 0}\) satisfying
\begin{equation}{\label{kareq3438}}\tag{29}
E\left [\int_{0}^{t}(Y_{s}^{(j)})^{2}ds\right ];\mbox{ \(1\leq j\leq n\)}
\end{equation}
for every \(t>0\), and
\begin{equation}{\label{kareq3439}}\tag{30}
M_{t}=\sum_{j=1}^{n}\int_{0}^{t}Y_{s}^{(j)}dW_{s}^{(j)}\mbox{ for }t\geq 0.
\end{equation}
In particular, \(M\) is a.s. continuous. Furthermore , if \(\widetilde{Y}^{(j)}\) for \(1\leq j\leq n\) are any other progressively measurable processes satisfying \((\ref{kareq3438})\) and \((\ref{kareq3439})\), then
\[\sum_{j=1}^{n}\int_{0}^{\infty}\left |Y_{t}^{(j)}-\widetilde{Y}_{t}^{(j)}\right |^{2}dt=0\mbox{ a.s.} \sharp\]
Theorem. (Representation of Brownian, Square-Integrable Martingales as Stochastic Integrals). Let \(\{({\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a \(d\)-dimensional Brownian motion as in Theorem \ref{kart3415}. Let \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) be a local martingale with \(M_{0}=0\) and RCLL paths a.s. Then there exist progressively measurable processes \(\{(Y_{t}^{(j)},{\cal F}_{t})\}_{t\geq 0}\) such that
\[\int_{0}^{t}(Y_{s}^{(j)})^{2}ds<\infty\mbox{ for }1\leq j\leq n,t\geq 0,\]
and
\[M_{t}=\sum_{j=1}^{n}\int_{0}^{t}Y_{s}^{(j)}dW_{s}^{(j)}\mbox{ for }t\geq 0.\]
In particular, \(M\) is a.s. continuous. \(\sharp\)
\begin{equation}{\label{karp3417}}\tag{31}\mbox{}\end{equation}
Proposition \ref{karp3417}. Under the hypotheses of Theorem \ref{kart3415} and with \(t>0\), let \(\xi\) be an \({\cal F}_{t}\)-measurable random variable with \(\mathbb{E}[\xi^{2}]<\infty\). Then there exist progressively measurable processes \(Y^{(1)},\cdots ,Y^{(n)}\) satisfying \((\ref{kareq3438})\) and such that
\[\xi =E[\xi]+\sum_{j=1}^{n}\int_{0}^{t}Y_{s}^{(j)}dW_{s}^{(j)}\mbox{ \(\mathbb{P}\)-a.s.} \sharp\]
We extend Proposition \ref{karp3417} to include the case \(t=\infty\). Recall that for \(M\in {\cal M}_{2}^{c}\), we denote by \({\cal L}_{\infty}^{*}(M)\) the class of processes \(X\) which are progressively measurable with respect to the filtration of \(M\) and which satisfy \(\mathbb{E}\left [\int_{0}^{\infty}X_{t}^{2}d\langle M\rangle_{t}\right ]<\infty\). When \(X\in {\cal L}_{\infty}^{*}(M)\), we have \(\int_{0}^{\infty}X_{t}dM_{t}\) defined \(P\)-a.s. If \({\bf W}\) is a \(n\)-dimensional Brownian motion, we denote by \({\cal L}_{\infty}^{*}({\bf W})\) the set of processes \(X\) which are progressively measurable with respect to the (augmented) filtration of \({\bf W}\) and which satisfy \(\mathbb{E}\left [\int_{0}^{\infty}X_{t}^{2}dt\right ]<\infty\).
Proposition. Under the hypotheses of Theorem \ref{kart3415}, let \(\xi\) be an \({\cal F}_{\infty}\)-measurable random variable with \(E[\xi^{2}]<\infty\). Then there are processes \(Y^{(1)},\cdots ,Y^{(n)}\) in
${\cal L}_{\infty}^{*}({\bf W})$ such that
\[\xi =E[\xi ]+\sum_{j=1}^{n}\int_{0}^{\infty}Y_{s}^{(j)}dW_{s}^{(j)}\mbox{ \(\mathbb{P}\)-a.s.} \sharp\]
Proposition. Let \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) be a continuous local martingale and assume that \(\lim_{t\rightarrow\infty}\langle M\rangle_{t}=\infty\) \(\mathbb{P}\)-a.s. Define the stopping time
$T(s)=\inf\{t\geq 0:\langle M\rangle_{t}>s\}$ for each \(s\geq 0\) and let \(W\) be the one-dimensional Brownian motion \(\{(W_{s}\equiv M_{T(s)},{\cal E}_{s})\}_{s\geq 0}\) as in Theorem~\ref{kart346} except now we take the filtration \(\{{\cal E}_{s}\}_{s\geq 0}\) to be the augmentation with respect to \(P\) of the filtration \(\{{\cal F}_{s}^{W}\}_{s\geq 0}\) generated by \(W\). Then, for every \({\cal F}_{\infty}\)-measurable random variable \(\xi\) satisfying \(\mathbb{E}[\xi^{2}]<\infty\), there is a process \(X\in {\cal L}_{\infty}^{*}(M)\) for which
\[\xi =E[\xi ]+\int_{0}^{\infty}X_{t}dM_{t}\mbox{ \(\mathbb{P}\)-a.s.}\sharp\]
Proposition. Let \(\{(W_{t},{\cal F}_{t})\}_{t\geq 0}\) be a standard one-dimensional Brownian motion, where, in addition to satisfying the usual conditions, \(\{{\cal F}_{t}\}_{t\geq 0}\) is left-continuous. If \(t>0\) and \(\xi\) is an \({\cal F}_{t}\)-measurable, a.s. finite random variable, then there exists a progressively measurable process \(\{(Y_{t},{\cal F}_{t})\}_{t\geq 0}\) satisfying \(\int_{0}^{t}Y_{s}^{2}dt<\infty\) \(\mathbb{P}\)-a.s. such that \(\xi =\int_{0}^{t}Y_{s}dW_{s}\) \(\mathbb{P}\)-a.s. \(\sharp\)
