Elizabeth Jane Gardner (1837-1922) was an American painter.
We have defined a semimartingale as a “good integrator” previously. This led naturally to defining the stochstic integral as a limit of sums. To express an integral as a limit of sums requires some path smoothness of the integrands and we limited the attention to processes in \({\bf L}\), the space of adapted processes with paths that are left-continuous and have right limits. The space \({\bf L}\) is sufficient to prove Ito’s formula, the Girsanov-Meyer theorem, and it also suffices in some applications such as stochastic differential equation. But other uses, such as martingale representation theory or local times, requires a large space of integrands. Here we define stochastic integration for predictable processes. The extension is very roughly analogous to how the Lebesgue integral extends the Riemann integral. We first define stochastic integration for bounded, predictable processes and a subclass of semimartingales known as \({\cal H}^{2}\). We then extend the definition to arbitrary semimartingales and to locally bounded predictable integrands. The topics are
- Stochastic Integration for Predictable Integrands \ref{a}
- Martingale Representation
- Stochastic Integration Depending on a Parameter
- Local Times
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Stochastic Integration for Predictable Integrands.
Here we will weaken the restriction that an integrand \(H\) must be in \({\bf L}\); we will show the definition of stochastic integrals can be extended to a class of predictably measurable integrands. Now \(X\) will denote a semimartingale with \(X_{0}=0\). This is a convenience involving no loss of generality: If \(Y\) is any semimartingale we can set \(\hat{Y}_{t}=Y_{t}-Y_{0}\), and if we have defined stochastic integrals
for semimartingales that are zero at \(0\), we can next define
\[\int_{0}^{t}H_{s}dY_{s}\equiv\int_{0}^{t}H_{s}d\hat{Y}_{s}+H_{0}Y_{0}.\]
When \(Y_{0}\neq 0\), recall that we write \(\int_{0+}^{t} H_{s}dY_{s}\) to denote integration on \((0,t]\), and \(\int_{0}^{t}H_{s}dY_{s}\) denotes integration on the closed interval \([0,t]\).
Let \(X=M+A\) be a decomposition of a semimartingale \(X\) with \(X_{0}=M_{0}=A_{0}=0\). Here \(M\) is a local martingale and \(A\) is an FV process (such a decomposition exists by Theorem \ref{prot322}). We will first consider special semimartingales. Recall that a semimartingale \(X\) is called special if it has a decomposition \(X=\bar{N}+\bar{A}\), where \(\bar{N}\) is a local martingale and \(\bar{A}\) is a locally natural (or equivalently predictable) FV process. This decomposition is unique by Proposition \ref{prot318} and it is called the canonical decomposition.
Definition. Let \(X\) be a special semimartingale with canonical decomposition \(X=\bar{N}+\bar{A}\). The \({\cal H}^{2}\) norm of \(X\) is defined to be
\[\parallel X\parallel_{{\cal H}^{2}}=\parallel\langle\bar{N},\bar{N}\rangle_{\infty}^{1/2}\parallel_{L^{2}}+\left |\!\left |\int_{0}^{\infty}|d\bar{A}_{s}|\right |\!\right |_{L^{2}}.\]
The space \({\cal H}^{2}\) of semimartingales consists of all special semimartingales with finite \({\cal H}^{2}\) norm. \(\sharp\)
Theorem. (Protter \cite{pro}) The space of \({\cal H}^{2}\) semimartingale is a Banach space.
Definition. The predictable \(\sigma\)-algebra \({\cal P}\) on \(\mathbb{R}_{+}\times \Omega\) is the smallest \(\sigma\)-algebra making all processes in \({\bf L}\) measurable. That is, \(\mathbb{P}=\sigma\{H:H\in {\bf L}\}\). We let \({\bf b}\mathbb{P}\) denote bounded processes that are \(\mathbb{P}\)-measurable. \(\sharp\)
We recall that \({\bf bL}\) denotes the space of adapted processes with bounded LCRL paths. The results that follow will enable us to extend the class of stochastic integrands from \({\bf bL}\) to \({\bf b}\mathbb{P} \)latex with \(X\in {\cal H}^{2}\) (and \(X_{0}=0\)). First we observe that if \(H\in {\bf bL}\) and \(X\in {\cal H}^{2}\), then the stochastic integral \(H\bullet X\in {\cal H}^{2}\). Also if \(X=\bar{N}+\bar{A}\) is the canonical decomposition of \(X\), then \(H\bullet\bar{N}+H\bullet\bar{A}\) is the canonical decomposition of \(H\bullet X\). Moreover,
\[\parallel H\bullet X\parallel_{{\cal H}^{2}}=\left |\!\left |\left (\int_{0}^{\infty}H_{s}^{2}d\langle\bar{N},\bar{N}\rangle_{s}\right )^{\frac{1}{2}}
\right |\!\right |_{L^{2}}+\left |\!\left |\int_{0}^{\infty}|H_{s}||d\bar{A}_{s}|\right |\!\right |_{L^{2}}.\]
The key idea in extending the integral is to notice that \(\langle\bar{N},\bar{N}\rangle\) and \(\bar{A}\) are FV processes, and therefore \(\omega\)-by-$\omega$ the integrals
\[\int_{0}^{t}H_{s}^{2}(\omega )d\langle\bar{N},\bar{N}\rangle_{s}(\omega )\mbox{ and }\int_{0}^{t}|H_{s}||d\bar{A}_{s}|\]
make sense for any \(H\in {\bf b}{\cal P}\) and not just \(H\in {\bf L}\).
Definition. Let \(X\in {\cal H}^{2}\) with \(X=\bar{N}+\bar{A}\) its canonical decomposition, and let \(H,J\in {\bf b}{\cal P}\). We define \(d_{X}(H,J)\) by
\[d_{X}(H,J)=\left |\!\left |\left (\int_{0}^{\infty}(H_{s}-J_{s})^{2}d\langle\bar{N},\bar{N}\rangle_{s}\right )^{\frac{1}{2}}\right |\!\right |_{L^{2}}+\left |\!\left |
\int_{0}^{\infty}|H_{s}-J_{s}||d\bar{A}_{s}|\right |\!\right |_{L^{2}}.\]
Theorem. (Protter \cite{pro}). For \(X\in {\cal H}^{2}\) the space \({\bf bL}\) is dense in \({\bf b}{\cal P}\) under \(d_{X}(\cdot ,\cdot )\).
Proposition (Protter \cite{pro}). Let \(X\in {\cal H}^{2}\) and \(H^{n}\in {\bf bL}\) such that \(H^{n}\) is Cauchy under \(d_{X}\). Then \(H^{n}\bullet X\) is Cauchy in \({\cal H}^{2}\).
Proof.
Since
\[\parallel H^{n}\bullet X-H^{m}\bullet X\parallel_{{\cal H}^{2}}=d_{X}(H^{n},H^{m}),\]
the result is immediate. \(\blacksquare\)
Proposition. (Protter \cite{pro}). Let \(X\in {\cal H}^{2}\) and \(H\in {\bf b}{\cal P}\). Suppose \(H^{n}\in {\bf bL}\) and \(J^{m}\in {\bf bL}\) are two sequences such that
\[\lim_{n\rightarrow\infty} d_{X}(H^{n},H)=\lim_{m\rightarrow\infty}d_{X}(J^{m},H)=0.\]
Then \(H^{n}\bullet X\) and \(J^{m}\bullet X\) tend to the same limit in \({\cal H}^{2}\).
Proof.
Let
\[Y=\lim_{n\rightarrow\infty}H^{n}\bullet X\mbox{ and }Z=\lim_{m\rightarrow\infty}J^{m}\bullet X,\]
where the limits are taken in \({\cal H}^{2}\). For \(\epsilon >0\), by taking \(n\) and \(m\) large enough, we have
\begin{align*}
\parallel Y-Z\parallel_{{\cal H}^{2}} & \leq\parallel Y-H^{n}\bullet X\parallel_{{\cal H}^{2}}+\parallel H^{n}\bullet X-J^{m}\bullet X\parallel_{{\cal H}^{2}}+\parallel J^{m}\bullet X-Z\parallel_{{\cal H}^{2}}\\
& \leq 2\epsilon +\parallel H^{n}\bullet X-J^{m}\bullet X\parallel_{{\cal H}^{2}}\\
& \leq 2\epsilon +d_{X}(H^{n},J^{m})\\
& \leq 2\epsilon +d_{X}(H^{n},H)+d_{X}(H,J^{m})\\
& \leq 4\epsilon ,
\end{align*}
and the result follows. \(\blacksquare\)
we are now in a position to define the stochastic integral for \(H\in {\bf b}\mathbb{P}\) and \(X\in {\cal H}^{2}\).
Definition. Let \(X\) be a semimartingale in \({\cal H}^{2}\) and let \(H\in {\bf b}{\cal P}\). Let \(H^{n}\in {\bf bL}\) be such that \(\lim_{n\rightarrow\infty}d_{X}(H^{n},H)=0\). The stochastic integral \(H\bullet X\) is the \((\)unique$)$ semimartingale \(Y\in {\cal H}^{2}\) such that \(\lim_{n\rightarrow\infty}H^{n}\bullet X=Y\) in \({\cal H}^{2}\). We write
\[ H\bullet X=\left\{\int_{0}^{t}H_{s}dX_{s}\right\}_{t\geq 0}.\]
We have defined the stochastic integral for predictable integrands and semimartingales in \({\cal H}^{2}\) as limits of the (previously defined) stochastic integrals. In order to investigate the properties of this more
general integral, we need to have approximations converging uniformly. The next proposition and its corollary give us this.
Proposition. (Protter \cite{pro}). Let \(X\) be a semimartingale in \({\cal H}^{2}\). Then
\[\mathbb{E}\left [(\sup_{t}|X_{t}|)^{2}\right ]\leq 8\cdot\parallel X\parallel^{2}_{{\cal H}^{2}}.\]
\begin{equation}{\label{proc45}}\tag{1}\mbox{}\end{equation}
Corollary \ref{proc45}. Let \(\{X^{n}\}\) be a sequence of semimartingales converging to \(X\) in \({\cal H}^{2}\). Then there exists a subsequence \(\{n_{k}\}\) such that
\[\lim_{n_{k}\rightarrow\infty} (X^{n_{k}}-X)^{*}=0\mbox{ a.s.}\]
We next investigate some of the properties of this generalized stochastic integral. Almost all of the properties established before still hold.
Proposition. Let \(X,Y\in {\cal H}^{2}\) and \(H,K\in {\bf b}{\cal P}\). Then
\[(H+K)\bullet X=H\bullet X+K\bullet X\mbox{ and }H\bullet (X+Y)=H\bullet X+H\bullet Y.\]
Proof. One need only check that it is possible to take a sequence \(H^{n}\in {\bf bL}\) that approximates \(H\) in both \(d_{X}\) and \(d_{Y}\). \(\blacksquare\)
Proposition. Let \(T\) be a stopping time. Then
\[(H\bullet X)^{T}=(H\cdot I_{[0,T]})\bullet X=H\bullet (X^{T}).\]
Proof. Note that \(I_{[0,T]}\in {\bf bL}\), so \(H\cdot I_{[0,T]}\in {\bf b}{\cal P}\). Also, \(X^{T}\) is clearly still in \({\cal H}^{2}\). Since we know this result is true for \(H\in {\bf bL}\) in Proposition \ref{prot212}, the result follows by uniform approximation using Corollary \ref{proc45}. \(\blacksquare\)
Proposition. The jump process \(\{\Delta (H\bullet X)\}_{s\geq 0}\) is indistinguishable from \(\{H_{s}\bullet (\Delta X_{s})\}_{s\geq 0}\).
\begin{equation}{\label{proc48}}\tag{2}\mbox{}\end{equation}
Corollary \refl{proc48}. Let \(X\in {\cal H}^{2}\), \(H\in {\bf b}{\cal P}\), and \(T\) a finite stopping time. Then \(H\bullet (X^{T-})=(H\bullet X)^{T-}\).
Proposition. Let \(X\in {\cal H}^{2}\) have paths of finite variation on compacts, and \(H\in {\bf b}{\cal P}\), Then \(H\bullet X\) agrees with a path-by-path Lebesgue-Steiltjes inetgral.
Proposition. (Associativity). Let \(X\in {\cal H}^{2}\) and \(H,K\in {\bf b}{\cal P}\). Then \(K\bullet X\in {\cal H}^{2}\) and \(H\bullet (K\bullet X)=(HK)\bullet X\).
Proposition. Let \(X\in {\cal H}^{2}\) be a \((\)square integrable$)$ martingale, and \(H\in {\bf b}{\cal P}\). Then \(H\bullet X\) is a square integrable martingale.
Proposition. Let \(X,Y\in {\cal H}^{2}\) and \(H,K\in {\bf b}{\cal P}\). Then
\[\langle H\bullet X,K\bullet Y\rangle_{t}=\int_{0}^{t}H_{s}K_{s}d\langle X,Y\rangle_{s}\mbox{ for }t\geq 0,\]
and in particular
\[\langle H\bullet X,H\bullet X\rangle_{t}=\int_{0}^{t}H_{s}^{2}d\langle X,X\rangle_{s}\mbox{ for }t\geq 0.\]
The restrictions of integrands to \({\bf b}{\cal P}\) and semimartingales to \({\cal H}^{2}\) are mathematically convenient but not necessary. A standard method of relaxing such hypothesis is to consider cases where they hold locally. Recall that {\bf a property \(\pi\) is said to hold locally} for a process \(X\) if there exists a sequence of stopping times \(\{T^{n}\}_{n\geq 0}\) such that \(0=T^{0}\leq T^{1}\leq T^{2}\leq\cdots\leq T^{n}\leq\cdots\) and \(\lim_{n\rightarrow\infty} T^{n}=\infty\) a,s,, and such that \(X^{T^{n}}\cdot I_{\{T^{n}>0\}}\) has property \(\pi\) for each \(n\). Since we are assuming the semimartingales \(X\) satisfy \(X_{0}=0\), we could as well require only that \(X^{T^{n}}\) has property \(\pi\) for each \(n\). A related condition is that a property hold prelocally.
Definition. A property \(\pi\) is said to hold {\bf prelocally} for a process \(X\) with \(X_{0}=0\) if there exists a sequence of stopping times \(\{T^{n}\}_{n\geq 1}\) increasing to \(\infty\) a.s. such that \(X^{T^{n}-}\) has property \(\pi\) for each \(n\geq 1\). \(\sharp\)
Recall that
\[X^{T-}=X_{t}\cdot I_{\{0\leq t<T\}}+X_{T-}\cdot I_{\{t\geq T\}}.\]
The next result shows that the restriction of semimartingales to \({\cal H}^{2}\) is not really a restriction at all.
\begin{equation}{\label{prot413}}\tag{3}\mbox{}\end{equation}
Proposition \ref{prot413}. (Protter \cite{pro}). Let \(X\) be a semimartingale with \(X_{0}=0\). Then \(X\) is pre-locally in \({\cal H}^{2}\). That is, there exists a nondecreasing sequence of stopping times \(\{T^{n}\}\), \(\lim_{n\rightarrow\infty}T^{n}=\infty\) a.s. such that \(X^{T^{n}-}\in {\cal H}^{2}\) for each \(n\). \(\sharp\)
We are now in a position to define the stochastic integral for an arbitrary semimartingales, as well as for predictable processes which need not be bounded. Let \(X\) be a semimartingale in \({\cal H}^{2}\). To define a stochastic integral for predictable processes \(H\) which are not necessarily bounded (written as \(H\in {\cal P}\)), we approximate then with \(H^{n}\in {\bf b}{\cal P}\).
Definition. Let \(X\in {\cal H}^{2}\) with canonical decomposition \(X=\bar{N}+\bar{A}\). We say \(H\in {\cal P}\) is \(({\cal H}^{2},X)\)-integrable if
\[\mathbb{E}\left [\int_{0}^{\infty}H_{s}^{2}d\langle\bar{N},\bar{N}\rangle_{s}\right ]+\mathbb{E}\left [\left (\int_{0}^{\infty}|H_{s}||d\bar{A}_{s}|\right )^{2}\right ]<\infty.\]
\begin{equation}{\label{prot414}}\tag{4}\mbox{}\end{equation}
Proposition \refl{prot414}. (Protter \cite{pro}). Let \(X\) be a semimartingale and let \(H\in {\cal P}\) be \(({\cal H}^{2},X)\)-integrable. Let \(H^{n}=H\cdot I_{\{|H|\leq n\}}\in {\bf b}{\cal P}\). Then \(H^{n}\bullet X\) is a Cauchy sequence in \({\cal H}^{2}\).
Proof. Since \(H^{n}\in {\bf b}{\cal P}\) for each \(n\), the stochastic integrals \(H^{n}\bullet X\) are defined. Note also that \(\lim_{n\rightarrow\infty}H^{n}=H\) and that \(|H^{n}|\leq |H|\) for each \(n\). Then
\[\parallel H^{n}\bullet X-H^{m}\bullet X\parallel_{{\cal H}^{2}}=d_{X}(H^{n},H^{m})=\left |\!\left |\left (\int_{0}^{\infty}
(H_{s}^{n}-H_{s}^{m})d\langle\bar{N},\bar{N}\rangle_{s}\right )^{1/2}\right |\!\right |_{{\cal H}^{2}}+\left |\!\left |\int_{0}^{\infty}|H_{s}^{n}-H_{s}^{m}||d\bar{A}_{s}|\right |\!\right |_{{\cal H}^{2}},\]
and the result follows by two applications of the dominated convergence theorem of Lebesgue. \(\blacksquare\)
Definition. Let \(X\) be a semimartingale in \({\cal H}^{2}\), and let \(H\in {\cal P}\) be \(({\cal H}^{2},X)\)-integrable. The stochastic integral \(H\bullet X\) is defined to be \(\lim_{n\rightarrow\infty}H^{n}\bullet X\) with convergence in \({\cal H}^{2}\), where \(H^{n}=H\cdot I_{\{|H|\leq n\}}\). \(\sharp\)
Note that \(H\bullet X\) in the preceding definition exists by Proposition \ref{prot414}. We “localize” the above proposition by allowing both more general \(H\in {\cal P}\) and arbitrary semimartingales
with the next definition.
Definition. Let \(X\) be a semimartingale and \(H\in {\cal P}\). The stochastic integral \(H\bullet X\) is said to exist if there exixts a sequence of stopping times \(\{T^{n}\}\) increasing to \(\infty\) a.s. such that
$X^{T^{n}-}\in {\cal H}^{2}$ for each \(n\geq 1\), and such that \(H\) is \(({\cal H}^{2},X^{T^{n}-})\)-integrable for each \(n\). In this case, we say \(H\) is \(X\)-integrable, weitten as \(H\in L(X)\), and we define the stochastic integral by
\[ H\bullet X=H\bullet (X^{T^{n}-})\mbox{ on }[0,T^{n})\]
for each \(n\). \(\sharp\)
Note that if \(m>n\) then
\[ H^{k}\bullet (X^{T^{m}-})^{T^{n}-}=H^{k}\bullet (X^{T^{m}\wedge T^{n}-})=H^{k}\bullet (X^{T^{n}-}),\]
where \(H^{k}=H\cdot I_{\{|H|\leq k\}}\), by Corollary~\ref{proc48}. Hence taking limits we have \(H\bullet (X^{T^{m}-})^{T^{n}-}=H\bullet (X^{T^{n}-})\), and the stochastic integral is well-defined for \(H\in L(X)\). Moreover let \(\{S^{m}\}\) be another sequence of stopping times such that \(X^{S^{m}-}\in {\cal H}^{2}\) and such that \(H\) is \(({\cal H}^{2},X^{S^{m}-})\)-inetgrable for each \(m\). Again using
Corollary \ref{proc48} combined with taking limits we see that \(H\bullet (X^{S^{m}-})=H\bullet (X^{T^{n}-})\) on \([0,S^{m}\wedge T^{n})\) for each \(n\geq 1\) and \(m\geq 1\). Thus in this sense the definition of the stochastic integral does not depend on the particular sequence of stopping times. If \(H\in {\bf b}{\cal P}\) (i.e. \(H\) is bounded), then \(H\in L(X)\) for all semimartingales \(X\), since every semimartingale is prelocally in \({\cal H}^{2}\) by Proposition \ref{prot413}.
Definition. A process \(H\) is said to be {\bf locally bounded} if there exists a sequence of stopping times \(\{S^{m}\}_{m\geq 1}\) increasing to \(\infty\) a.s. such that \(\left\{H_{t\wedge S^{m}}\cdot I_{\{S^{m}>0\}}\right\}_{t\geq 0}\) is bounded for each \(m\geq 1\). \(\sharp\)
Note that any process in \({\bf L}\) is locally bounded.
Proposition. (Protter \cite{pro}). Let \(X\) be a semimartingale and let \(H\in \mathbb{P}\) be locally bounded. Then \(H\in L(X)\). That is, the stochastic integral \(H\bullet X\) exists. \(\sharp\)
We now turn the attention to the properties of this more general integral. Many of the properties are simple extensions of earlier results. Note that trivially the stochastic integral \(H\bullet X\) for \(H\in L(X)\) is also a semimartingale.
Proposition. (Protter \cite{pro}). Let \(X\) be a semimartingale and let \(H,J\in L(X)\). Then \(\alpha H+\beta J\in L(X)\) and \((\alpha H+\beta J)\bullet X=\alpha H\bullet X+\beta J\bullet X\). That is, \(L(X)\) is a linear space. \(\sharp\)
Proposition. (Protter \cite{pro}). Let \(X\) and \(Y\) be semimartingales and suppoe \(H\in L(X)\) and \(H\in L(Y)\). Then \(H\in L(X+Y)\) and \(H\bullet (X+Y)=H\bullet X+H\bullet Y\). \(\sharp\)
Proposition. Let \(X\) be a semimartingale and \(H\in L(X)\). The jump process \(\{\Delta (H\bullet X)_{s}\}_{s\geq 0}\) is indistinguishable from \(\{H_{s}\bullet (\Delta X_{s})\}_{s\geq 0}\). \(\sharp\)
Proposition. Let \(T\) be a stopping time, \(X\) a semimartingale, and \(H\in L(X)\). Then
\[(H\bullet X)^{T}=H\cdot I_{[0,T]}\bullet X=H\bullet (X^{T}).\]
Letting \(\infty -\) equal \(\infty\), we have moreover
\[(H\bullet X)^{T-}=H\bullet (X^{T-}).\]
Proposition. Let \(X\) be a semimartingale with paths of finite variation on compacts. Let \(H\in L(X)\) be such that the Stieltjes integral \(\int_{0}^{t}|H_{s}||dX_{s}|\) exists a.s. for each \(t\geq 0\). Then the stochastic integral \(H\bullet X\) agrees with a path-by-path Stieltjes integral. \(\sharp\)
Proposition. Let \(X\) be a semimartingale with \(K\in L(X)\). Then \(H\in L(K\bullet X)\) if and only if \(HK\in L(X)\), in which case \(H\bullet (K\bullet X)=(HK)\bullet X\). \(\sharp\)
Proposition. Let \(X\) and \(Y\) be semimartingales and let \(H\in L(X)\) and \(K\in L(Y)\). Then
\[\langle H\bullet X,K\bullet Y\rangle_{t}=\int_{0}^{t}H_{s}K_{s}d\langle X,Y\rangle_{s}\]
for each \(t\geq 0\). \(\sharp\).
Proposition. Let \(X\) be a semimartingale, let \(H\in L(X)\), and suppose \(\mathbb{Q}\) is another probability measure with \(\mathbb{Q}\ll\mathbb{P}\). If \(H_{\mathbb{Q}}\bullet X\) exists
($H_{\mathbb{Q}}\bullet X$ denotes the stochastic integral computed under \(\mathbb{Q}\)), it is
$\mathbb{Q}$-indistinguishable from \(H_{\mathbb{P}}\bullet X\). \(\sharp\)
Proposition. Let \(X\) be a semimartingale and \(H\in L(X)\). If \(\mathbb{Q}\ll \mathbb{P}\) then \(H\in L(X)\) under \(\mathbb{Q}\) as well and \(H_{\mathbb{Q}}\bullet X=X_{\mathbb{P}}\bullet X\) \(\mathbb{Q}\)-a.s. \(\sharp\).
\begin{equation}{\label{prot426}}\tag{5}\mbox{}\end{equation}
Proposition \ref{prot426}. Let \(X\) and \(\bar{X}\) be two semimartingales, and let \(H\in L(X)\), \(\bar{H}\in L(\bar{X})\). Let
\[A=\{\omega :H_{\cdot}(\omega )=\bar{H}_{\cdot}(\omega )\mbox{ and }X_{\cdot}(\omega )=\bar{X}_{\cdot}(\omega )\},\]
and let
\[B=\{\omega :t\mapsto X_{t}(\omega )\mbox{ is of finite variation on compacts}\}.\]
Then \(H\bullet X=\bar{H}\bullet\bar{X}\) on \(A\), and \(H\bullet X\) is equal to a path by path Lebesgue-Steiltjes integral on \(B\). \(\sharp\)
Corollary. With the notation of Proposition~\ref{prot426}, let \(S\) and \(T\) be two stopping times with \(S<T\). Define
\[C=\{\omega :H_{t}(\omega )=\bar{H}_{t}(\omega ), X_{t}(\omega )=\bar{X}_{t}(\omega ), S(\omega )<t\leq T(\omega )\}\]
and
\[D=\{\omega :t\mapsto X_{t}(\omega )\mbox{ is of finite variation on }S(\omega )<t<T(\omega )\}.\]
Then \(H\bullet X^{T}-H\bullet X^{S}=\bar{H}\bullet\bar{X}^{T}-\bar{H}\bullet\bar{X}^{S}\) on \(C\) and \(H\bullet X^{T}-H\bullet X^{S}\) equals a path-by-path Lebesgue-Steiltjes integral on \(D\). \(\sharp\)
Proposition. Let \(M\) be a locally square integrable local martingale, and let \(H\in {\cal P}\). The stochastic integral \(H\bullet M\) exists \((\)i.e. \(H\in L(M))\) and is a locally square integrable local martingale if there exists a sequence of stopping times \(\{T^{n}\}_{n\geq 1}\) increasing to \(\infty\) a.s. such that
\[E\left [\int_{0}^{T^{n}}H_{s}^{2}d\langle M,M\rangle_{s}\right ]<\infty .\sharp\]
Proposition. Let \(M\) be a local martingale, and let \(H\in {\cal P}\) be locally bounded. Then the stochastic integral \(H\bullet M\) is a local martingale. \(\sharp\)
Corollary. Let \(M\) be a local martingale with \(M_{0}=0\), and let \(T\) be a predictable stopping time. Then \(M^{T-}\) is a local martingale. \(\sharp\)
Proposition. Let \(M\) be a continuous local martingale and let \(H\in {\cal P}\) be such that \(\int_{0}^{t}H_{s}^{2}d\langle M,M\rangle_{s}<\infty\) a.s. for each \(t\geq 0\). Then the stochastic integral \(H\bullet M\) exists \((\)i.e. \(H\in L(M))\) and it is a continuous local martingale. \(\sharp\)
Corollary. Let \(X\) be a continuous semimartingale with \((\)unique$)$ decomposition \(X=M+A\). Let \(H\in {\cal P}\) be such that
\[\int_{0}^{t}H_{s}^{2}d\langle M,M\rangle_{s}+\int_{0}^{t}|H_{s}||dA_{s}|<\infty\mbox{ a,s.}\]
for each \(t\geq 0\). Then the stochastic integral \((H\bullet X)_{t}=\int_{0}^{t}H_{s}dX_{s}\) exists and it is continuous. \(\sharp\)
In the preceding corollary the semimartingale \(X\) is continuous, hence \(\langle X,X\rangle =\langle M,M\rangle\) and the hypothesis can be written equivalently as
\[\int_{0}^{t}H_{s}^{2}d\langle X,X\rangle_{s}+\int_{0}^{t}|H_{s}||dA_{s}|<\infty\mbox{ a,s.}\]
for each \(t\geq 0\).
Proposition. Let \(M\) be a local martingale with jumps bounded by a constant \(\beta\). Let \(H\in {\cal P}\) be such that \(\int_{0}^{t}H_{s}^{2}d\langle M,M\rangle_{s}<\infty\) a.s. for \(t\geq 0\), and \(E[H_{T}^{2}]<\infty\) for any bounded stopping time \(T\). Then, the stochastic integral \(\{\int_{0}^{t}H_{s}dM_{s}\}_{t\geq 0}\) exists and it is a local martingale. \(\sharp\)
Theorem. (Dominated Convergence Theorem). Let \(X\) be a semimartingale, and let \(H^{m}\in {\cal P}\) be a sequence converging to a limt \(H\) a.s. If there exists a process \(G\in L(X)\) such that \(|H^{m}|\leq G\) for all \(m\), then \(H^{m}\) and \(H\) are in \(L(X)\) and \(H^{m}\bullet X\) converges to \(H\bullet X\) in ucp. \(\sharp\)
Proposition. Let \(\{{\cal F}_{t}\}_{t\geq 0}\) and \(\{{\cal G}_{t}\}_{t\geq 0}\) be two filtrations satisfying the usual conditions and suppose \({\cal F}_{t}\subset {\cal G}_{t}\) for each \(t\geq 0\), and that \(X\) is a semimartingale for both \(\{{\cal F}_{t}\}_{t\geq 0}\) and \(\{{\cal G}_{t}\}_{t\geq 0}\). Let \(H\) be locally bounded and predictable for \(\{{\cal F}_{t}\}_{t\geq 0}\). Then, the stochastic integrals \(H_{{\cal F}}\bullet X\) and \(H_{{\cal G}}\bullet X\) both exists, and they are equal, where \(H_{{\cal F}}\bullet X\) and \(H_{{\cal G}}\bullet X\) denote the stochastic integrals computed with the filtrations
$\{{\cal F}_{t}\}_{t\geq 0}$ and \(\{{\cal G}_{t}\}_{t\geq 0}\), respectively. \(\sharp\)
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Martingale Representation.
Now we will be concerned with martingales, rather than semimartingales. The question of martingale representation is the following: given a collection \({\cal A}\) of martingales (or local martingales), when can all
martingales (or all local martingales) be represented as stochastic integrals with respect to processes in \({\cal A}\)? We assume as given an underlying complete, filtered probability space $latex (\Omega ,{\cal F},
\{{\cal F}_{t}\}_{t\geq 0})$ satisfying the usual conditions.
Definition. The space \({\bf M}^{2}\) of square integrable martingales is all martingales \(M\) such that
\[\sup_{t}\mathbb{E}[M_{t}^{2}]<\infty\mbox{ and }M_{0}=0\mbox{ a.s.}\]
Notice that if \(M\in {\bf M}^{2}\), then
\[\lim_{t\rightarrow\infty}\mathbb{E}[M_{t}^{2}]=\mathbb{E}[M_{\infty}^{2}]<\infty\mbox{ and }M_{t}=\mathbb{E}[M_{\infty}|{\cal F}_{t}].\]
Thus each \(M\in {\bf M}^{2}\) can be identified with its terminal value \(M_{\infty}\). We can endow \({\bf M}^{2}\) with a norm
\[\parallel M\parallel =(\mathbb{E}[M_{\infty}^{2}])^{1/2}=(\mathbb{E}[[M,M]_{\infty}])^{1/2},\]
and also with an inner product
\[(M,N)=\mathbb{E}[M_{\infty}N_{\infty}]\]
for \(M,N\in {\bf M}^{2}\). It is evident that \({\bf M}^{2}\) is a Hilbert space and that its dual space is also \({\bf M}^{2}\). \(\sharp\)
Recall that \(M_{t}^{T}=M_{t\wedge T}\).
Definition. A closed subspace \({\bf F}\) of \({\bf M}^{2}\) is called a stable subspace if it is stable under stopping; that is, if \(M\in {\bf F}\) and if \(T\) is a stopping time, then \(M^{T}\in {\bf F}\). \(\sharp\)
Proposition. (Protter \cite{pro}). Let \({\bf F}\) be a closed subspace of \({\bf M}^{2}\). Then the following statements are equivalent
(a) \({\bf F}\) is closed under the operation: For \(M\in {\bf F}\), \((M-M^{t})\cdot I_{\Gamma}\in {\bf F}\) for \(\Gamma\in {\cal F}_{t}\), any \(t\geq 0\);
(b) \({\bf F}\) is a stable subspace;
(c) if \(M\in {\bf F}\) and \(H\) is bounded and predictable, then \(\{\int_{0}^{t}H_{s}dM_{s}\}_{t\geq 0}=H\bullet M\in {\bf F}\);
(d) if \(M\in {\bf F}\) and \(H\) is predictable with \(\mathbb{E}[\int_{0}^{\infty}H_{s}^{2}d\langle M,M\rangle_{s}]<\infty\), then \(H\bullet M\in {\bf F}\). \(\sharp\)
Since the arbitrary intersection of closed, stable subspace is still closed and stable, we can make the following definition.
Definition. Let \({\cal A}\) be a subset of \({\bf M}^{2}\). The stable subspace generated by \({\cal A}\), denoted by \(S({\cal A})\), is the intersection of all closed, stable subspaces containing \({\cal A}\). \(\sharp\)
We can identify a martingale \(M\in {\bf M}^{2}\) with its terminal value \(M_{\infty}\in L^{2}\). Therefore another martingale \(N\in {\bf M}^{2}\) is orthogonal to \(M\) if \(E[N_{\infty}M_{\infty}]=0\).
Definition. Two martingales \(N,M\in {\bf M}^{2}\) are said to be strongly orthogonal if their product \(L=NM\) is a \((\)uniformly integrable$)$ martingale. \(\sharp\)
Note that if \(N,M\in {\bf M}^{2}\) are strongly orthogonal, then \(NM\) a (uniformly integrable) martingale implies that \([N,M]\) is also a local martingale; it is a uniformly integrable martingale by the Kunita-Watanabe inequality. Thus \(M,N\in {\bf M}^{2}\) are strongly orthogonal if and only if \([M,N]\) is a uniformly integrable martingale. If \(N\) and \(M\) are strongly orthogonal than \(\mathbb{E}[N_{\infty}M_{\infty}]=\mathbb{E}[L_{\infty}]=\mathbb{E}[L_{0}]=0\), so strong orthogonality implies orthogonality. The converse is not true however.
Definition. For a subset \({\cal A}\) of \({\bf M}^{2}\), we let \({\cal A}^{\perp}\) \((\)resp. \({\cal A}^{\times})\) denote the set of all elements of \({\bf M}^{2}\) orthogonal \((\)resp. strongly orthogonal$)$ to each element of \({\cal A}\). \(\sharp\)
Proposition. If \({\cal A}\) is any subset of \({\bf M}^{2}\), then \({\cal A}^{\times}\) is \((\)closed and$)$ stable. \(\sharp\)
Proposition. Let \(M,N\in {\bf M}^{2}\). The the following statements are equivalent
(i) \(M\) and \(N\) are strongly orthogonal;
(ii) \(S(\{M\})\) and \(N\) are strongly orthogonal;
(iii) \(S(\{M\})\) and \(S(\{N\})\) are strongly orthogonal;
(iv) \(S(\{M\})\) and \(N\) are orthogonal;
(v) \(S(\{M\})\) and \(S(\{N\})\) are orthogonal. \(\sharp\)
\begin{equation}{\label{prot435}}\tag{6}\mbox{}\end{equation}
Proposition \ref{prot435}. Let \(M^{1},\cdots ,M^{n}\in {\bf M}^{2}\), and suppose \(M^{i},M^{j}\) are strongly orthogonal for \(i\neq j\). Then \(S(\{M^{1},\cdots ,M^{n}\})\) consists of the set of stochastic integrals
\[ H^{1}\bullet M^{1}+\cdots +H^{n}\bullet M^{n}=\sum_{i=1}^{n}H^{i}\bullet M^{i},\]
where \(H^{i}\) is predictable and
\[\mathbb{E}\left [\int_{0}^{\infty}(H_{s}^{i})^{2}d\langle M^{i},M^{i}\rangle_{s}\right ]<\infty\mbox{ for }1\leq i\leq n.\]
\end{Pro}
\begin{equation}{\label{prot436}}\tag{7}\mbox{}\end{equation}
Proposition \ref{prot436}. Let \({\cal A}\) be a subset of \({\bf M}^{2}\) which is stable. Then \({\cal A}^{\perp}\) is a stable subspace, and if \(M\in {\cal A}^{\perp}\) then \(M\) is strongly orthogonal to \({\cal A}\). That is, \({\cal A}^{\perp} ={\cal A}^{\times}\) and \(S({\cal A})=A^{\perp\perp}={\cal A}^{\times\perp}={\cal A}^{\times\times}\). \(\sharp\)
\begin{equation}{\label{proc436}}\tag{8}\mbox{}\end{equation}
Corollary \ref{proc436}. Let \({\cal A}\) be a stable subspace of \({\bf M}^{2}\). Then, each \(M\in {\bf M}^{2}\) has a decomposition \(M=A+B\) with \(A\in {\cal A}\) and \(B\in {\cal A}^{\times}\).
Proof. \({\cal A}\) is a closed subspace of \({\bf M}^{2}\), so each \(M\in {\bf M}^{2}\) has a decomposition into \(M=A+B\) with \(A\in {\cal A}\) and \(B\in {\cal A}^{\perp}\). However \({\cal A}^{\perp}={\cal A}^{\times}\) by Proposition \ref{prot436}. \(\blacksquare\)
Corollary. Let \(M,N\in {\bf M}^{2}\), and let \(L\) be the projection of \(N\) onto \(S(\{M\})\), the stable subspace generated by \(\{M\}\). Then there exists a predictable process \(H\) such that \(L=H\bullet M\).
Proof. We know that such an \(L\) exists by Corollary~\ref{proc436}. Since \(\{M\}\) consists of just one element we can apply Proposition~\ref{prot435} to obtain the result. \(\blacksquare\)
Definition. Let \({\cal A}\) be finite set of martingales in \({\bf M}^{2}\). We say that \({\cal A}\) has the \((\)predictable$)$ representation property if \({\cal I}={\bf M}^{2}\), where
\[{\cal I}=\left\{X:X=\sum_{i=1}^{n}H^{i}\bullet M^{i},M^{i}\in {\cal A}\right\},\]
each \(H^{i}\) predictable such that
\[E\left [\int_{0}^{\infty}(H_{s}^{i})^{2}d\langle M^{i},M^{i}\rangle_{s}\right ]<\infty\mbox{ for }1\leq i\leq n.\sharp\]
Corollary. Let \({\cal A}=\{M^{1},\cdots ,M^{n}\}\subset {\bf M}^{2}\), and suppose \(M^{i},M^{j}\) are strongly orthogonal for \(i\neq j\). Suppose further that if \(N\in {\bf M}^{2}\), \(N\perp {\cal A}\) in the strong sense implies that \(N=0\). Then \({\cal A}\) has the predictable representation property.
Proof. By Proposition \ref{prot435}, we have \(S({\cal A})={\cal I}\). The hypotheses imply that \(S({\cal A})^{\perp}=\{0\}\), hence \(S({\cal A})={\bf M}^{2}\). \(\blacksquare\)
Definition. Let \({\cal A}\subset {\bf M}^{2}\). The set of \({\bf M}^{2}\) martingale measure for \({\cal A}\), denoted by \({\cal M}^{2}({\cal A})\), is the set of all probability measures \(Q\) defined on \(\wedge_{0\leq t<\infty} {\cal F}_{t}\) such that
(a) \(Q\ll P\);
(b) \(Q=P\) on \({\cal F}_{0}\);
(c)}if \(X\in {\cal A}\) then \(X\in {\bf M}^{2}(Q)\),
where \({\bf M}^{2}(Q)\) denotes all \((Q,\{{\cal F}_{t}\}_{t\geq 0})\) square integrable martingales. \(\sharp\)
Proposition. The set \({\cal M}^{2}({\cal A})\) is a convex set.
Proof. Let \(Q,R\in {\cal M}^{2}({\cal A})\), and let \(S=\lambda Q+(1-\lambda )R\) for \(0<\lambda <1\). Then for \(X\in {\cal M}({\cal A})\),
\[\sup_{t}\mathbb{E}_{S}[M_{t}^{2}]=\sup_{t}\left (\lambda\cdot \mathbb{E}_{Q}[M_{t}^{2}]+(1-\lambda )\cdot\mathbb{E}_{R}[M_{t}^{2}]\right )<\infty,\]
since \(Q,R\in {\cal M}^{2}({\cal A})\). Also if \(H\in b{\cal F}_{s}\) for \(s<t\), then
\[\mathbb{E}_{S}[M_{t}H]=\lambda\mathbb{E}_{Q}[M_{t}H]+(1-\lambda )\mathbb{E}_{R}[M_{t}H]=\lambda E_{Q}[M_{s}H]+(1-\lambda )\mathbb{E}_{R}[M_{s}H]=\mathbb{E}_{S}[M_{S}H],\]
and \(S\in {\cal M}^{2}({\cal A})\). \(\blacksquare\)
Definition. A measure \(Q\in {\cal M}({\cal A})\) is an extremal point of \({\cal M}^{2}({\cal A})\) if whenever \(Q=\lambda R+(1-\lambda )S\) with \(R,S\in {\cal M}^{2}({\cal A})\), \(R\neq S\), \(0\leq\lambda\leq 1\), then \(\lambda =0\) or \(1\). \(\sharp\)
Proposition. Let \({\cal A}\subset {\bf M}^{2}\). If \(S({\cal A})={\bf M}^{2}\) then \(P\) is an extremal point of \({\cal M}^{2}({\cal A})\).
Proof. Suppose \(P\) is not extremal. We will show that \(S({\cal A})\neq {\bf M}^{2}\). Since \(P\) is not extremal, there exist \(Q,R\in {\cal M}^{2}({\cal A})\), \(Q\neq R\), such that \(P=\lambda Q+(1-\lambda )R\) for \(0<\lambda <1\). Let
\[L_{\infty}=\frac{dQ}{dP}\mbox{ and }L_{t}=E\left [\left .\frac{dQ}{dP}\right |{\cal F}_{t}\right ].\]
Then
\[1=\frac{dP}{dP}=\lambda\cdot L_{\infty}+(1-\lambda )\frac{dP}{dP}\geq\lambda\cdot L_{\infty}\mbox{ a.s.},\]
hence \(L_{\infty}\leq 1/\lambda\) a.s. Therefore \(L\) is a bounded martingale with \(L_{0}=1\) (since \(Q=P\) on \({\cal F}_{0}\)), and thus \(L-L_{0}\) is a nonconstant martingale in \({\bf M}^{2}(P)\). However if \(X\in {\cal A}\) and \(H\in b{\cal F}_{s}\), then \(X\) is a \(Q\)-martingale and for \(s<t\)
\[\mathbb{E}_{P}[X_{t}L_{t}H]=\mathbb{E}_{P}[X_{t}L_{\infty}H]=\mathbb{E}_{Q}[X_{t}H]=\mathbb{E}_{Q}[X_{s}H]=\mathbb{E}_{P}[X_{s}L_{\infty}H]=\mathbb{E}_{P}[X_{s}L_{s}H],\]
and \(XL\) is a \(P\)-martingale. Therefore \(X(L-L_{0})\) is a \(P\)-martingale, and \(L-L_{0}\in {\bf M}^{2}\) and it is strongly orthogonal to \({\cal A}\). By Proposition\ref{prot436}, we cannot have \(S({\cal A})={\bf M}^{2}\). \(\blacksquare\)
Proposition. Let \({\cal A}\subset {\bf M}^{2}\). If \(P\) is an extremal point of \({\cal M}^{2}({\cal A})\), then every bounded \(P\)-martingale strongly orthogonal to \({\cal A}\) is null.
Proof. Let \(L\) be a bounded nonconstant martingale strongly orthogonal to \({\cal A}\). Let \(c\) be a bound for \(|L|\), and set
\[dQ=\left (1-\frac{L_{\infty}}{2c}\right )dP\mbox{ and }dR=\left (1+\frac{L_{\infty}}{2c}\right )dP.\]
We have \(Q,R\in {\cal M}^{2}({\cal A})\), and \(P=\frac{1}{2}Q+\frac{1}{2}R\) is a decomposition that shows that \(P\) is not extremal, a contradiction. \(\blacksquare\)
Proposition. Let \({\cal A}=\{M^{1},\cdots ,M^{n}\}\subset {\bf M}^{2}\) with \(M^{i}\) continuous and \(M^{i},M^{j}\) strongly orthogonal for \(i\neq j\). Suppose that \(P\) is an extremal point of \({\cal M}^{2}({\cal A})\). Then, we have the following properties
(i) every stopping time is accessible;
(ii) every bounded martingale is continuous;
(iii) every uniformly integrable martingale is continuous;
(iv) \({\cal A}\) has the predictable representation property. \(\sharp\)
Proposition. Let \(M\in {\bf M}^{2}\), \(Y^{n}\in {\bf M}^{2}\) for \(n\geq 1\) and suppose \(Y_{\infty}^{n}\) converges to \(Y_{\infty}\) in \(L^{2}\) and that there exists a sequence \(H^{n}\in L(M)\) such that \(Y_{t}^{n}=\int_{0}^{t}H_{s}^{n} dM_{s}\) for \(n\geq 1\). Then there exists a predictable process \(H\) in \(L(M)\) such that \(Y_{t}=\int_{0}^{t}H_{s}dM_{s}\).
Proof. If \(Y_{\infty}^{n}\) converges to \(Y_{\infty}\) in \(L^{2}\), then \(Y^{n}\) converges to \(Y\) in \({\bf M}^{2}\). By Proposition~\ref{prot435} we have that \(S(\{M\})={\cal I}(\{M\})\), the stochastic integrals with respect to \(M\). Moreover \(Y^{n}\in S(\{M\})\) for each \(n\). Therefore \(Y\) is in the closure of \(S(\{M\})\); but \(S(\{M\})\) is closed, so \(Y\in S(\{M\})={\cal I}(\{M\})\), and the proof is complete. \(\sharp\)
Proposition. Let \({\cal A}=\{M^{1},M^{2},\cdots ,M^{n},\cdots\}\) with \(M^{i}\in {\bf M}^{2}\), and suppose there exist disjoint predictable sets \(\Gamma_{i}\) such that \(I_{\Gamma^{i}}d\langle M^{i},M^{i}\rangle =d\langle M^{i}.M^{i}\rangle\) for \(i\geq 1\). Let
\[A_{t}=\sum_{i=1}^{\infty}\int_{0}^{t}I_{\Gamma^{i}}d\langle M^{i},M^{i}\rangle_{s}.\]
Suppose that \(\mathbb{E}[A_{\infty}]<\infty\), and that , for \({\cal F}^{i}\subset {\cal F}_{\infty}\) such that for any \(X^{i}\in b{\cal F}^{i}\), we have $latex X_{t}^{i}=\mathbb{E}[X^{i}|{\cal F}_{t}]=
\int_{0}^{t}H_{s}^{i}dM_{s}^{i}$, \(t\geq 0\) for some predictable process \(H^{i}\). Then \(M=\sum_{i=1}^{\infty} M^{i}\) exists and is in \({\bf M}^{2}\), and for any \(Y\in b\wedge_{i}{\cal F}_{i}\), if \(Y_{t}=\mathbb{E}[Y|{\cal F}_{t}]\), we have that \(Y_{t}=\int_{0}^{t}H_{s}dM_{s}\) for the martingale \(M=\sum_{i=1}^{\infty}M^{i}\) and for some \(H\in L(M)\). \(\sharp\)
Proposition. Let \({\bf X}=(X^{1},\cdots ,X^{n})\) be an \(n\)-dimensional Brownian motion and let \(\{{\cal F}_{t}\}_{t\geq 0\leq\infty}\) denote its completed natural filtration. Then every locally square integrable local martimgale \(M\) for \(\{{\cal F}_{t}\}_{t\geq 0}\) has a representation
\[M_{t}=M_{0}+\sum_{i=1}^{n}\int_{0}^{t}H_{s}^{i}dX_{s}^{i},\]
where \(H^{i}\) is \((\)predictable, and$)$ in \(L(X^{i})\).
\end{Pro}
\begin{equation}{\label{proc442}}\tag{9}\mbox{}\end{equation}
Corollary \ref {proc442}. Let \(\{{\cal F}_{t}\}_{t\geq 0}\) be the completed natural filtration of an \(n\)-dimensional Brownian motion. Then every local martingale \(M\) for \(\{{\cal F}_{t}\}_{t\geq 0}\) is continuous. \(\sharp\)
Corollary. Let \({\bf X}=(X^{1},\cdots ,X^{n})\) be an \(n\)-dimensional Brownian motion and let \(\{{\cal F}_{t}\}_{0\leq t\leq\infty}\) be its completed natural filtration. Then every local martingale \(M\) for \(\{{\cal F}_{t}\}_{t\geq 0}\) has a representation
\[M_{t}=M_{0}+\sum_{i=1}^{n}\int_{0}^{t}H_{s}^{i}dX_{s}^{i}\]
where \(H^{i}\) are predictable. \(\sharp\)
Corollary. Let \({\bf X}=(X^{1},\cdots ,X^{n})\) be an \(n\)-dimensional Brownian motion and let \(\{{\cal F}_{t}\}_{0\leq t\leq \infty}\) be its completed natural filtration. Let \(Z\in {\cal F}_{\infty}\) be in \(L^{1}\). Then there exist \(H^{i}\) predictable in \(L(X^{i})\) with \(\int_{0}^{\infty}(H_{s}^{i})^{2}ds<\infty\) a.s. such that
\[Z=E[Z]+\sum_{i=1}^{n}\int_{0}^{\infty}H_{s}^{i}dX_{s}^{i}.\sharp\]
Corollary. Let \({\bf X}=(X^{1},\cdots ,X^{n})\) be an \(n\)-dimensional Brownian motion and let \(\{{\cal F}_{t}\}_{0\leq t\leq\infty}\) be its completed natural filtration. Let \(Z\in L^{1}({\cal F}_{\infty})\) and \(Z>0\) a.s. Then there exist \(J^{i}\) predictable with \(\int_{0}^{\infty} (J_{s}^{i})^{2}ds<\infty\) a.s. such that
\[Z=E[Z]\cdot\exp\left (\sum_{i=1}^{n}\int_{0}^{\infty}J_{s}^{i}dX_{s}^{i}-\frac{1}{2}\int_{0}^{\infty} (J_{s}^{i})^{2}ds\right ).\sharp\]
Corollary. Let \(\{{\cal F}_{t}\}_{t\geq 0}\) be the completed natural filtration of an \(n\)-dimensional Brownian motion. If \(T\) is a totally inaccessible stopping time, then \(T=\infty\) a.s. \(\sharp\)
\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}
Stochastic Integration Depending on a Parameter.
$(A,{\cal A})$ denotes a measurable space.
Proposition. Let \(\{Y^{n}(a,t,\omega )\}\) be a sequence of process such that \({\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) is measurable, and that, for each fixed \(a\), the process \(Y^{n}(a,t,\omega )\) is RCLL. Suppose \(Y^{n}(a,t,\cdot )\) converges ucp for each \(a\in A\). Then, there exixts an \({\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) measurable process \(Y=Y(a,t,\omega )\) such that \(Y(a,t,\cdot )= \lim_{n\rightarrow\infty}Y^{n}(a,t,\cdot )\) with convergence in ucp, and \(Y\) is a.s. RCLL for each \(a\in A\). \(\sharp\)
\begin{equation}{\label{prot444}}\tag{10}\mbox{}\end{equation}
Proposition \ref{prot444}. Let \(X\) be a semimartingale with \(X_{0}=0\) a.s. and let \(H(a,t,\omega ) \equiv H_{t}^{a}(\omega )\) be \({\cal A}\otimes {\cal P}\) measurable and bounded, where \({\cal P}\)
is the predictable \(\sigma\)-algebra. Then there is a function \(Z(a,t,\omega )\in {\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) such that for each \(a\in A\), \(Z(a,t,\omega )\) is a RCLL, adapted version of the stochastic integral \(\int_{0}^{t}H_{s}^{a}dX_{s}\). \(\sharp\)
\begin{equation}{\label{proc444}}\tag{i1}\mbox{}\end{equation}
Corollary \ref{proc444}. Let \(X\) be a semimartingale with \(X_{0}=0\) a.s., and let \(H(a,t,\omega )= H_{t}^{a}(\omega )\in {\cal A}\times {\cal P}\) be such that for each \(a\) the process $latex H^{a}\in
L(X)$. Then there exists a function \(Z(a,t,\omega )= Z_{t}^{a}\in {\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) such that for each \(a\), \(Z_{t}^{a}\) is an a.s. RCLL version of \(\int_{0}^{t}H_{s}^{a}dX_{s}\). \(\sharp\)
Theorem (Fubini’s Theorem). Let \(X\) be a semimartingale, \(H_{t}^{a}=H(a,t,\omega )\) be a bounded \({\cal A}\otimes {\cal P}\) measurable fuction, and let \(\mu\) be a finite measure on \({\cal A}\). Let \(Z_{t}^{a}=\int_{0}^{t}H_{s}^{a}dX_{s}\) be \({\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) measurable such that for each \(a\), \(Z^{a}\) is a RCLL version of \(H^{a}\bullet X\). Then \(Y_{t}=\int_{A}Z_{t}^{a}\mu (da)\) is a RCLL version of \(H\bullet X\), where \(H_{t}=\int_{A}H_{t}^{a}\mu (da)\). \(\sharp\)
\begin{equation}{\label{{prot446}}\tag{12}\mbox{}\end{equation}
Theorem \ref{prot446}. (Fubini’s Theorem; Second Version). Let \(X\) be a semimartingale, let \(H_{t}^{a}=H(a,t,\omega )\) be \({\cal A}\otimes {\cal P}\) measurable, let \(\mu\) be a finite positive measure on \(A\), and assume
\[\left (\int_{A}(H_{t}^{a})^{2}\mu (da)\right )^{1/2}\in L(X).\]
Letting \(Z_{t}^{a}=\int_{0}^{t}H_{s}^{a}dX_{s}\) be \({\cal A}\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\) measurable and \(Z^{a}\) RCLL for each \(a\), then \(Y_{t}=\int_{A}Z_{t}^{a}\mu (da)\) exists and is RCLL version of \(H\bullet X\), where \(H_{t}=\int_{A}H_{t}^{a} \mu (da)\). \(\sharp\)
The hypotheses of Proposition~\ref{prot446} are slightly unnatural, since they are not invariant under the transformation
\[ H\rightarrow\frac{1}{\phi (a)}H^{a}\mbox{ and }\mu\rightarrow\phi (a)\mu (da)\]
where \(\phi\) is any positive function such that \(\int\phi (a)\mu (da)<\infty\). This can be alleviated by replacing the assumption
\[\left (\int_{A}(H^{a})^{2}\mu (da)\right )^{1/2}\in L(X)\mbox{ with }\left (\int\frac{(H^{a})^{2}}{\phi (a)}\mu (da)\right )^{1/2}\in L(X)\]
for some positive \(\phi\in L^{1}(d\mu )\). One can also relax the assumption on \(\mu\) to be \(\sigma\)-finite rather than finite.
\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}
Local Times.
(Oksendal \cite{oks}, P.58, Ex4.10) What happens if we try to apply the Ito formula t \(g(W_{t})\) when \(W_{t}\) is one-dimensional and \(g(x)=|x|\)? In this case, \(g\) is not \(C^{2}\) at \(x=0\), so we modify \(g(x)\) near \(x=0\) to \(g_{\epsilon}(x)\) as follows
\[g_{\epsilon}(x)=\left\{\begin{array}{ll}
|x| & \mbox{if \(|x|\geq\epsilon\)}\\
\frac{1}{2}\left (\epsilon +\frac{x^{2}}{\epsilon}\right ) &
\mbox{if \(|x|<\epsilon\)}
\end{array}\right .\]
where \(\epsilon >0\). Then we can show that
\[g_{\epsilon}(W_{t})=g_{\epsilon}(W_{0})+\int_{0}^{t}g’_{\epsilon}(W_{s})dW_{s}+\frac{1}{2\epsilon}\cdot\mu (\{s\in [0,t]:W_{s}\in (-\epsilon ,\epsilon )\}),\]
where \(\mu (\cdot )\) denotes the Lebesgue measure. We also have
\[\int_{0}^{t}g’_{\epsilon}(W_{s})\cdot I_{\{W_{s}\in (-\epsilon ,\epsilon )\}}dW_{s}=\int_{0}^{t}\frac{W_{s}}{\epsilon}\cdot I_{\{W_{s}\in (-\epsilon ,\epsilon )\}}dW_{s}\rightarrow 0\]
in \(L^{2}(P)\) as \(\epsilon\rightarrow 0\). Furthermore by letting \(\epsilon\rightarrow 0\), we have that
\begin{equation}{\label{okseq4312}}
|W_{t}|=|W_{0}|+\int_{0}^{t}sign(W_{s})dW_{s}+L_{t}(\omega ),
\end{equation}
where
\[L_{t}=\lim_{\epsilon\rightarrow 0}\frac{1}{2\epsilon}\cdot\mu (\{s\in [0,t]:W_{s}\in (-\epsilon ,\epsilon )\})\mbox{ limit in \(L^{2}(P)\)}\]
and
\[sign(x)=\left\{\begin{array}{ll}
+1 & \mbox{if \(x\geq 0\)}\\ -1 & \mbox{if \(x<0\)}.
\end{array}\right .\]
$L_{t}$ is called the {\bf local time} for Brownian motion at \(0\) and (\ref{okseq4312}) is the Tanaka formula for Bronwian motion.
We have established Ito’s formula which showed that if \(f:\mathbb{R}\rightarrow \mathbb{R}\) is \(C^{2}\) and \(X\) is a semimartingale, then \(f(X)\) is again a semimartingale. That is, semimartingales are
preserved under \(C^{2}\) transformations. This property extends slightly: semimartingales are preserved under convex transformations, as Proposition \ref{prot447} below shows. Local times for semimartingales appear in the extension of Ito’s formula from \(C^{2}\) functions to convex functions (Theorem \ref{prot451})
\begin{equation}{\label{prot447}}\tag{13}\mbox{}\end{equation}
Proposition \ref{prot447}. Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be convex and let \(X\) be a semimartingale. Then \(f(X)\) is a semimartingale and one has
\[f(X_{t})-f(X_{0})=\int_{0+}^{t}f'(X_{s-})dX_{s}+A_{t}\]
where \(f’\) is the left-derivative of \(f\) and \(A\) is an adapted, right-continuous, increasing process. Moreover \(\Delta A_{t}=f(X_{t})-f(X_{t-})-f'(X_{t-})\Delta X_{t}\). \(\sharp\)
We recall that \(x^{+}=\max\{x,0\}\), \(x^{-}=-\min\{x,0\}\), \(x\vee y=\max\{x,y\}\) and \(x\wedge y=\min\{x,y\}\).
\begin{equation}{\label{proc447}}\tag{14}\mbox{}\end{equation}
Corollary \ref{proc447}} (Protter \cite{pro}). Let \(X\) be a semimartingale. Then \(|X|,X^{+},X^{-}\) are all semimartingales.
Proof. The functions \(f(x)=|x|\), \(g(x)=x^{+}\) and \(h(x)=x^{-}\) are all convex. The result follows by Proposition \ref{prot447}. \(\blacksquare\)
Corollary. (Protter \cite{pro}).
Let \(X\) and \(Y\) be semimartingales. Then \(X\vee Y\) and \(X\wedge Y\) are semimartingales.
Proof. Since semimartingales form a vector space and
\[x\vee y=\frac{1}{2}(|x-y|+x+y)\mbox{ and }x\wedge y=\frac{1}{2}(x+y-|x-y|),\]
the result is an immediate consequence of Corollary \ref{proc447}. \(\blacksquare\)
Proposition. The space of semimartingales is a vector space, an algebra, a lattice, and is stable under \(C^{2}\), and more generally under convex transformations. \(\sharp\)
Definition. Th sign function is defined to be
\[sign(x)=\left\{\begin{array}{ll}
1 & \mbox{if \(x>0\)}\\
-1 & \mbox{if \(x\leq 0\)}.
\end{array}\right .\]
We further define
\begin{equation}{\label{proeq4*}}
\left\{\begin{array}{l}
h_{0}(x)=|x|\\
h_{a}(x)=|x-a|.
\end{array}\right .
\end{equation}
The \(sign(x)\) is the left-derivative of \(h_{0}(x)\) and \(sign(x-a)\) is the left-derivative of \(h_{a}(x)\). Since \(h_{a}(x)\) is convex, by Proposition \ref{prot447}, we have for a semimartingale \(X\)
\begin{equation}{\label{proeq4**}}
h_{a}(X_{t})=|X_{t}-a|=|X_{0}-a|+\int_{0+}^{t}sign(X_{s-}-a)dX_{s}+A_{t}^{a},
\end{equation}
where \(A_{t}^{a}\) is the increasing process of Proposition \ref{prot447}. Using (\ref{proeq4*}) and (\ref{proeq4**}) as defined above we can define the local time of an arbitrary semimartingale.
Definition. Let \(X\) be a semimartingale, and let \(h_{a}\) and \(A^{a}\) be as defined in \((\ref{proeq4*})\) and \((\ref{proeq4**})\) above. The local time at \(a\) of \(X\), denoted by \(L_{t}^{a}=L^{a}(X)_{t}\), is defined to be the process given by
\[L_{t}^{a}=A_{t}^{a}-\sum_{0<s\leq t}\left [H_{a}(X_{s})-h_{a}(X_{s-})-h’_{a}(X_{s-})\Delta X_{s}\right ].\sharp\]
Notice that by Corollary \ref{proc444}, the integral \(\int_{0+}^{t} sign(X_{s-})-s)dX_{s}\) in (\ref{proeq4**}) has a version which is jointly measurabel in \((a,t,\omega )\) and RCLL in \(t\). Therefore so also does
$\{A_{t}^{a}\}_{t\geq 0}$, and finally asl also does the local time \(L_{t}^{a}\). We always choose the jpintly measurable, RCLL version of the local time. We further observe that the jumps of the process \(A^{a}\)
defined in (\ref{proeq4**}) are precisely \(\sum_{s\leq t}\left [h_{a}(X_{s})-h_{a}(X_{s-})-h’_{a}(X_{s-})\Delta X_{s}\right ]\) (by Proposition \ref{prot447}), and therefore the local time \(\{L_{t}^{a}\}_{t\geq 0}\) is continuous in \(t\). Indeed, the local time \(L^{a}\) is the continuous part of the increasing process \(A^{a}\).
Proposition. Let \(X\) be a semimartingale and let \(L^{a}\) be its local time at \(a\). Then
\[(X_{t}-a)^{+}-(X_{0}-a)^{+}=\int_{0}^{t}I_{\{X_{s-}>a\}}dX_{s}+\sum_{0<s\leq t}I_{\{X_{s-}>a\}}\cdot (X_{s}-a)^{-}+\sum_{0<s\leq t}I_{\{X_{s-}\leq a\}}\cdot (X_{s}-a)^{+}+\frac{1}{2}L_{t}^{a}\]
and
\[(X_{t}-a)^{-}-(X_{0}-a)^{-}=-\int_{0}^{t}I_{\{X_{s-}\leq a\}}dX_{s}+\sum_{0<s\leq t}I_{\{X_{s-}>a\}}\cdot (X_{s}-a)^{-}+\sum_{0<s\leq t}I_{\{X_{s-}\leq a\}}\cdot (X_{s}-a)^{+}+\frac{1}{2}L_{t}^{a}.\sharp\]
Proposition. (Protter \cite{pro}). Let \(X\) be a semimartingale, and let \(L_{t}^{a}\) be its local time at the level \(a\) for each \(a\in\mathbb{R}\). For a.a, \(\omega\), the measure in \(t\), \(dL_{t}^{a}\), is
carried by the set \(\{s:X_{s-}(\omega )=X_{s}(\omega )=a\}\). \(\sharp\)
The next theorem gives a very satisfying generalization of Ito’s formula.
\begin{equation}{\label{prot451}}\tag{15}\mbox{}\end{equation}
Theorem \ref{prot451}. (Meyer-Ito Formula). If \(f\) is the difference of two convex functions, let \(f’\) be its left-derivative, and let \(\mu\) be the signed measure \((\)when restricted to compacts$)$ which is the second derivative of \(f\) in the generalized function sense. Then the following equation holds
\[f(X_{t})-f(X_{0})=\int_{0+}^{t}f'(X_{s-})dX_{s}+\sum_{0<s\leq t}\left [f(X_{s})-f(X_{s-})-f'(X_{s-})\Delta X_{s}\right ]+\frac{1}{2}\int_{-\infty}^{+\infty}\mu (da)L_{t}^{a},\]
where \(X\) is a semimartingale and \(L_{t}^{a}=L_{t}^{a}(X)\) is its local time at \(a\). \(\sharp\)
\begin{equation}{\label{proc4511}}\tag{16}\mbox{}\end{equation}
Corollary \ref{proc4511}. Let \(X\) be a semimartingale with local time \(\{L^{a}\}_{a\in \mathbb{R}}\). Let \(g\) be a bounded Borel measurable functin. Then a.s.
\[\int_{-\infty}^{+\infty}L_{t}^{a}g(a)da=\int_{0}^{t}g(X_{s-})d\langle X,X\rangle_{s}^{c}. \sharp\]
Corollary. Let \(X\) be a semimartingale with local time \(\{L^{a}\}_{a\in \mathbb{R}}\). Then
\[\langle X,X\rangle_{t}^{c}=\int_{-\infty}^{+\infty} L_{t}^{a}da. \sharp\]
\begin{equation}{\label{proc4513}}\tag{17}\mbox{}\end{equation}
Corollary \ref{proc4513}. {(Meyer-Tanaka Formula). Let \(X\) be a semimartingale with continuous paths. Then
\[|X_{t}|=|X_{0}|+\in_{0+}^{t}sign(X_{s})dX_{s}+L_{t}^{0}. \sharp\]
When \(X_{t}=W_{t}\), a standard Brownian motion, then the term \(M_{t}=\int_{0}^{t}sign(W_{s})dW_{s}\) is a continuous local martingale, and
\[\langle M,M\rangle_{t}=\int_{0}^{t}sign(W_{s})^{2}d\langle W,W\rangle_{s}=\langle W,W\rangle_{t}=t.\]
Therefore by Levy’s theorem in Proposition~\ref{prot238}, we know that \(M_{t}\) is another Brownian motion. We therefore have, where \(W_{0}=x\) for standard Brownian motion,
\begin{equation}{\label{proeq41}}
|W_{t}|=|W_{0}|+\beta_{t}+L_{t},
\end{equation}
where \(\beta_{t}=\int_{0}^{t}sign(W_{s})dW_{s}\) is a Brownian motion, and \(L_{t}\) is the local time fo \(W\) at zero. Formula (\ref{proeq41}) is known as Tanaka’s Formula.
Observe that if \(f(x)=|x|\), then \(f”(x)=2\delta (x)\), where \(\delta\) is the “delta function at zero”, which of cource is a generalized function, or “distribution”. Thus Corollaries \ref{proc4511} and \ref{proc4513} give the intuitive interpretation of local time as
\[L_{t}^{0}=\int_{0}^{t}\delta (X_{s})d\langle X,X\rangle_{s}\mbox{ and }L_{t}^{a}=\int_{0}^{t}\delta (X_{s}-a)d\langle X,X\rangle_{s}\]
for continuous semimartingales.
Proposition. Let \(X\) be a continuous local martingale with \(X_{0}=0\), and let \(0<\alpha <1/2\). Then \(Y_{t}=|X_{t}|^{\alpha}\) is not a semimartingale unless \(X\) is identically zero. \(\sharp\)
We next wish to determine when there exists a version of the local time \(L_{t}^{a}\) which is jointly continuous in \((a,t)\mapsto L_{t}^{a}\) a.s., or jointly right-continuous in \((a,t)\mapsto L_{t}^{a}\).
Hypothesis A. For the remainder of this section, we let \(X\) denote a semimartingale with the restriction that \(\sum_{0<s\leq t}|\Delta X_{s}|<\infty\) a.s. for each \(t>0\).
Observe that if \((\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)\) is a probability space where \(\{{\cal F}_{t}\}_{t\geq 0}\) is the completed minimal filtration of a Brownian motion \(W=\{W_{t}\}_{t\geq 0}\), then all semimartingales on this Brownian space verify Hypothesis A: Indeed, by Corollary \ref{proc442}, all the local martingales are continuous. Thus if \(X\) is a semimartingale, let \(X=M+A\) be a decomposition with \(M\) a local martingale and \(A\) a FV process. Then the jump processes \(\Delta X\) and \(\Delta A\) are equal, hence
\[\sum_{0<s\leq t}|\Delta X_{s}|=\sum_{0<s\leq t}|\Delta A_{s}|<\infty,\]
since \(A\) is of finite variation on compacts.
Let \(X\) be a semimartingale satisfying Hypothesis A, and let
\[J_{t}=\sum_{0\leq s\leq t}\Delta X_{s};\]
that is, \(J\) is the process which is the sum of the jumps. Because of the hypothesis \(J\) is an FV process and thus also a semimartingale. Moreover \(Y_{t}=X_{t}-J_{t}\) is a continuous semimartingale with \(Y_{0}=0\), and we let \(Y=M+A\) be its (unique) decomposition, where \(M\) is a continuous local martingale and \(A\) is a continuous FV process with \(M_{0}=A_{0}=0\). Such a process \(M\) is then uniquely determined and we can write \(M=X^{c}\), the continuous local martingale part of \(X\).
We assume given a semimartingale \(X\) satisfying Hypothesis A. If \(Z\) is any other semimartingale we write
\[\hat{Z}_{t}^{a}=\int_{0+}^{t}I_{\{X_{s-}>a\}}dZ_{s}\mbox{ for }a\in \mathbb{R}.\]
It is always assumed that we take \({\cal B}(\mathbb{R})\otimes {\cal B}(\mathbb{R}_{+})\otimes {\cal F}\)
measurable, RCLL version of \(Z^{a}\) (ref. Proposition \ref{prot444}).
Theorem. (Burkholder’s Inequality). Let \(X\) be a continuous local martingale with \(X_{0}=0\), \(2\leq p<\infty\), and \(T\) a finite stopping time. Then
\[\mathbb{E}[(X_{T}^{*})^{p}]\leq C_{p}\mathbb{E}\left [[\langle X,X\rangle_{t}^{p/2}]\right ]\]
where
\[C_{p}=\left [q^{p}\cdot\frac{p(p-1)}{2}\right ]^{p/2}\mbox{ and }\frac{1}{p}+\frac{1}{q}=1.\sharp\]
Actually much more is true. Indeed for any local martingale (continuous or not) it is known that there exist constants \(c_{p}\) and \(C_{p}\) such that for a finite stopping time \(T\)
\[\mathbb{E}\left [(X_{T}^{*})^{p}\right ]^{1/p}\leq c_{p}\cdot \mathbb{E}\left [\langle X,X\rangle_{t}^{p/2}\right ]^{1/p}\leq C_{p}\cdot \mathbb{E}\left [(X_{T}^{*})^{p}\right ]^{1/p}\]
for \(1\leq p<\infty\).
Proposition. Let \(X\) be a semimartingale satisfying Hypothesis A. There exists a version of \((\hat{X}^{c})_{t}^{a}\) such that \((a,t,\omega )\mapsto (\hat{X}^{c})_{t}^{a}(\omega )\) is $latex {\cal
B}(\mathbb{R})\otimes {\cal P}$ measurable, and everywhere jointly continuous in \((a,t)\). \(\sharp\)
Proposition. Let \(X\) be a semimartingale satisfying Hypothesis A. Then there exists a \({\cal B}(\mathbb{R})\otimes {\cal P}\) measurable version of \((a,t,\omega ) \mapsto L_{t}^{a}(\omega )\) which is everywhere jointly right-continuous in \(a\) and continuous in \(t\). Moreover a.s. the limits \(L_{t}^{a-}= \lim_{b\rightarrow a,b<a} L_{t}^{a}\) exist. \(\sharp\)
Recall that for a semimartingale \(X\) satisfying Hypothesis A, we let \(J_{t}=\sum_{0\leq s\leq t}\Delta X_{s}\), \(Y=X-J\) and \(Y=M+A\) is the (unique) decomposition of \(Y\) with \(A_{0}=0\).
\begin{equation}{\label{proc456}}\tag{18}\mbox{}\end{equation}
Corollary \ref{proc456}. Let \(X\) be a semimartingale satisfyin Hypothesis A. Let \(X=M+A+J\) be a decomposition with \(M\) and \(A\) continuous and \(J\) the jump process of \(X\), and let \(\{L_{t}^{a}\}_{t\geq 0}\) be its local time at the level \(a\). Then
\[L_{t}^{a}-L_{t}^{a-}=2\int_{0}^{t}I_{\{X_{s-}=a\}}dA_{s}=2\int_{0}^{t}I_{\{X_{s}=a\}}dA_{s}. \sharp\]
An example of a semimartingale satisfying Hypothesis A but having a discontinuous local time is \(X_{t}=|W_{t}|\) where \(W\) is standard Brownian motion with \(W_{0}=0\). Here \(L^{a}(X)=L^{a}(W)+L^{-a}(W)\) for \(a>0\), \(L^{0}(X)=L^{0}(W)\) and \(L^{a}(X)=0\) for \(a<0\).
Corollary. Let \(X\) be a semimartingale satisfying Hypothesis A. Let \(A\) be as in Corollary \ref{proc456}. The local time \((L_{t}^{a})\) is continuous in \(t\) and is continuous at \(a=a_{0}\) if and only if
\[\int_{0}^{\infty} I_{\{X_{s}=a_{0}\}}|dA_{s}|=0. \sharp\]
Observe that if \(X=W\), a Brownian motion (or any continuous local martingale), then \(A=0\) and the local time of \(X\) can be taken everywhere jointly continuous.
\begin{equation}{\label{proc4563}}\tag{19}\mbox{}\end{equation}
Corollary \ref{proc4563}. Let \(X\) be a semimartingale satisfying Hypothesis A. Then for every \((a,t)\) we have
\[L_{t}^{a}=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\int_{0}^{t}I_{\{a\leq X_{s}\leq a+\epsilon\}}d\langle X,X\rangle_{s}^{c}\mbox{ a.s. and }
L_{t}^{a-}=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\int_{0}^{t}I_{\{a-\epsilon\leq X_{s}\leq a\}}d\langle X,X\rangle_{s}^{c}\mbox{ a.s.}\sharp\]
The lack of symmetry in the above formulas stems from the definition of local time, where we defined \(sign(x)\) in an asymmetric way
\[sign(x)=\left\{\begin{array}{ll}
1 & \mbox{if \(x>0\)}\\
-1 & \mbox{if \(x\leq 0\)}
\end{array}\right .\]
A symmetrized result follows trivially
\[\frac{L_{t}^{a}+L_{t}^{a-}}{2}=\lim_{\epsilon\rightarrow 0}\frac{1}{2\epsilon}\int_{0}^{t}I_{\{|X_{s}-a|\leq\epsilon\}}d\langle X,X\rangle_{s}^{c}\mbox{ a.s.}\]
If \(X=W\), a Brownian motion, then Corollary~\ref{proc4563} becomes the classical result
\[L_{t}^{a}=\lim_{\epsilon\rightarrow 0}\frac{1}{2\epsilon}\int_{0}^{t}I_{(a-\epsilon ,a+\epsilon )}(W_{s})ds\mbox{ a,s,}\]
Local time have interesting properties when viewed as processes with “time” fixed, and the space variables “a” as the parameter.
The Meyer-Ito formula can be extended in a different direction, which gives rise to the Bouleau-Yor formula, allowing nonconvex functions of semimartingales. The key idea is that the function \(a\mapsto L_{U}^{a}\) induces a measure on \(\mathbb{R}\) if \(L\) is the local time of a semimartingale \(X\) satisfying Hypothesis A and \(U\) is a positive random variable.
Proposition. Let \(X\) be a semimartingale satisfying Hypothesis A, \(U\) a positive random variable, \(L^{a}\) the local times of \(X\). Then the operation
\[f\mapsto \sum_{i=1}^{n}f_{i}\cdot (L_{U}^{a_{i+1}}-L_{U}^{a_{i}}),\]
where \(f(x)=\sum f_{i}\cdot I_{(a_{i},a_{i+1}]}(x)\), can be extended uniquely to a vector measure on \({\cal B}(\mathbb{R})\) with values in \(L^{0}\). \(\sharp\)
Corollary. Let \(X\) be a semimartingale satisfying Hypothesis A, \(U\) a positive random variable, and \(X_{t}^{a}=\int_{0}^{t}I_{\{X_{s-}\leq a\}}dX_{s}\). Then \(d_{a}X_{U}^{a}\) can be defined as an \(L^{0}\)-valued measure, and for \(f\in b{\cal B}(\mathbb{R})\), we have
\[\int_{\mathbb{R}}f(a)d_{a}X_{U}^{a}=\int_{0+}^{U}f(X_{s-})dX_{s}. \sharp\]
Theorem. (Bouleau-Yor Formula). Let \(X\) be a semimartingale satisfying Hypothesis A, \(U\) a positive random variable, \(f\) a bounded, Borel function, and \(F(x)=\int_{0}^{x}f(u)du\). Then
\[F(X_{U})-F(X_{0})=\int_{0+}^{U}f(X_{s-})dX_{s}-\frac{1}{2}\int f(a)d_{a}L_{U}^{a}+\sum_{0<s\leq U}\left (F(X_{s})-F(X_{s-})-f(X_{s-})\Delta X_{s}\right ). \sharp\]
Combining the Bouleau-Yor formula with the Meyer-Ito formula, we obtain the following relationship
Corollary. Let \(X\) be a semimartingale satisfying Hypothesis A, and let \(T\) be a finite-valued random variable. Let \(L^{a}\) be the local time of \(X\) and let \(f\) be a \(C^{1}\) function. Then
\[\int_{-\infty}^{+\infty}f'(a)L_{T}^{a}da=-\int_{-\infty}^{+\infty} f(a)d_{a}L_{T}^{a}.\]
