Semimartingales And Decomposable Processes

Edwin Thomas Roberts (1840-1917) was a British painter.

The topics are

Doob-Meyer Decompositions
Quasimartingales
The Fundamental Theorem of Local Martingales
Classical Semimartingales
Natural Versus Predictable Processes

Historically the stochastic integral was first proposed for Brownian motion, then for continuous martingales, then for square-integrable martingales and finally for processes which can be written as the sum of a locally square integrable local martingale and an adapted, RCLL processes with paths of finite variation on compacts; that is, a decomposable process.

Definition. An adapted, RCLL process $A$ is a finite variation process ({\bf FV}) if almost surely the paths of $A$ are of finite variation on each compact interval of $[0,\infty )$. We write $latex \int_{0}^{\infty}
|dA_{s}|$ or $|A|_{\infty}$ for the random variable which is the total variation of the paths of $A$. $\sharp$

Definition. An adapted, RCLL process $X$ is decomposable if there exist processes $M$ and $A$ satisfying
\[X_{t}=X_{0}+M_{t}+A_{t}\]
with $M_{0}=A_{0}=0$, $M$ a locally square integrable local martingale, and $A$ an FV process. $\sharp$

Definition. An adapted, RCLL process $Y$ is a {\bf classical semimartingale} if there exist processes $N$ and $B$ with $N_{0}=B_{0}=0$ satisfying
\[Y_{t}=Y_{0}+N_{t}+B_{t}\]
where $N$ is a local martingale and $B$ is an FV process. $\sharp$

Clearly an FV process is decomposable, and both FV processes and decomposable processes are semimartigales. Now the goal is to show that a process $X$ is a classical semimartingale if and only if it is a semimartingale.

Proposition. (Protter \cite{pro}). Let $X$ be an adapted, RCLL process. The following are equivalent

(a) $X$ is a semimartingale;

(b) $X$ is decomposable;

(d) given $\beta >0$, there exist $M$ and $A$ with $M_{0}=A_{0}=0$, $M$ a local martingale with jumps bounded by $\beta$, $A$ an FV process such that $X_{t}=X_{0}+M_{t}+A_{t}$. $\sharp$

Definition. An FV process $A$ is of {\bf integrable variation} if $\mathbb{E}\left [\int_{0}^{\infty}|dA_{s}|\right ]<\infty$. The FV process $A$ is of locally integrable variation if there exists a sequence of stopping times $\{T_{n}\}_{n\in {\bf N}}$ increasing to $\infty$ a.s. satisfying $\mathbb{E}\left [\int_{0}^{T_{n}}|dA_{s}|\right ]<\infty$ for each $n$. $\sharp$

Definition. Let $A$ be an adapted FV process of integrable variation with $A_{0}=0$. Then $A$ is a {\bf natural process} if $\mathbb{E}\left [\langle M,A\rangle _{\infty}\right ]=0$ for all bounded martingales $M$. $\sharp$

Definition. Let $A$ be an adapted FV process of locally integrable variation with $A_{0}=0$. $A$ is a {\bf locally natural process} if $\mathbb{E}\left [\langle M,A^{T}\rangle _{\infty}\right ]=0$ for any stopping time $T$ such that $E\left [\int_{0}^{T}|dA_{s}|\right ]<\infty$, and for all bounded martingales $M$. $\sharp$

Proposition. Let $A$ be an FV process with $A_{0}=0$, and $\mathbb{E}[|A|_{\infty}]<\infty$. Then $A$ is natural if and only if
\[\mathbb{E}\left [\int_{0}^{\infty} M_{s-}dA_{s}\right ]=\mathbb{E}[M_{\infty}\cdot A_{\infty}]\]
for any bounded martingale $M$. $\sharp$

Note that if $A$ is an FV process with $A_{0}=0$ then it is a quadratic pure jump semimartingale, whence
\[\langle M,A\rangle _{\infty}=\sum_{0<s<\infty}\Delta M_{s}\cdot\Delta A_{s}.\]
In particular, if $A$ is continuous then $\langle M,A\rangle =0$, and $A$ is a locally natural process.

Proposition. Let $X$ be a semimartingale. If $X$ has a decomposition $X_{t}=X_{0}+M_{t}+A_{t}$ with $M$ a local martingale and $A$ a locally natural FV process with $M_{0}=A_{0}=0$, then such a decomposition is unique. $\sharp$

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

Doob-Meyer Decompositions.

Definition. An adapted, RCLL process $X$ is a {\bf potential} if it is a nonnegative supermartingale such that $\lim_{t\rightarrow\infty} E[X_{t}]=0$. A process $\{X_{n}\}_{n\in {\bf N}}$ is also called a potential if it is a nonnegative supermartingale for ${\bf N}$ and $\lim_{n\rightarrow\infty}\mathbb{E}[X_{n}]=0$. $\sharp$

Theorem. (The Doob Decomposition). A potential $\{X_{n}\}_{n\in {\bf N}}$ has a decomposition $X_{n}=M_{n}-A_{n}$, where $A_{n+1}\geq A_{n}$ a.s., $A_{0}=0$, $A_{n}\in {\cal F}_{n-1}$, and $M_{n}=\mathbb{E}[A_{\infty}|{\cal F}_{n}]$. Such a decomposition is unique.

Proof. Let $M_{0}=X_{0}$ and $A_{0}=0$. Define $M_{1}=M_{0}+(X_{1}-\mathbb{E}[X_{1}|{\cal F}_{0}])$ and $A_{1}=X_{0}-\mathbb{E}[X_{1}|{\cal F}_{0}]$. Define $M_{n}$ and $A_{n}$ inductively as follows
\[M_{n}=M_{n-1}+(X_{n}-E[X_{n}|{\cal F}_{n-1}])\mbox{ and }A_{n}=A_{n-1}+(X_{n-1}-E[X_{n}|{\cal F}_{n-1}]).\]
Note that $\mathbb{E}[A_{n}]=\mathbb{E}[X_{0}]-\mathbb{E}[X_{n}]\leq \mathbb{E}[X_{0}]<\infty$, as is easily checked by induction. It is then simple to check that $M_{n}$ and $A_{n}$ so defined satisfy the hypotheses. Next suppose $X_{n}=N_{n}-B_{n}$ is another representation. Then $M_{n}-N_{n}=A_{n}-B_{n}$ and in particular $M_{1}-N_{1}=A_{1}-B_{1}\in {\cal F}_{0}$; thus $latex M_{1}-N_{1}=
\mathbb{E}[M_{1}-N_{1}|{\cal F}_{0}]=M_{0}-N_{0}=X_{0}-X_{0}=0$, hence $M_{1}=N_{1}$. Continuing inductively shows $M_{n}=N_{n}$ for all $n$. $\blacksquare$

\begin{equation}{\label{prot35}}\tag{1}\mbox{}\end{equation}

Theorem \ref{prot35}. (The Riesz Decomposition). Let $\{X_{t}\}_{t\geq 0}$ be a positive supermartingale. Then there exists a unique decomposition of $X$ into a martingale and a potential $Z$ satisfying $X=M+Z$.

Proof. As is well known $\lim_{t\rightarrow\infty}X_{t}=Y$ a.s., and moreover $Y\in L^{1}$. Let $M_{t}=E[Y|{\cal F}_{t}]$, and let $Z_{t}=X_{t}-M_{t}$. One easily verifies that $Z$ is a potential. Let $X=N+W$ be another decomposition. Then $N-M=Z-W$ is a martingale with $Z_{\infty}-W_{\infty}=0$. Therefore $Z=W$ and $N=M$ since $Z_{t}-W_{t}=E[Z_{\infty}-W_{\infty}|{\cal F}_{t}]$. $\blacksquare$

\begin{equation}{\label{prot36}}\tag{}\mbox{}\end{equation}
2
Theorem \ref{prot36}. (The Doob-Meyer Decomposition). Let $X$ be a potential such that the collection ${\cal H}=\{X_{T}:\mbox{$T$ a stopping time}\}$ is uniformly integrable. Then $X$ has a
decomposition $X=M-A$, where $M$ is a uniformly integrable martingale and $A$ is a right-continuous, increasing process with $A_{0}=0$, and $A$ is natural. Such a decomposition is unique. $\sharp$

Corollary. (The Doob-Meyer Decomposition). Let $X$ be a positive supermartingale, and suppose ${\cal H}=\{X_{T}:\mbox{$T$ a stopping time}\}$ is uniformly integrable. Then $X$ has a unique decomposition $X=M-A$ where $M$ is a martingale and $A$ is a right-continuous, increasing and natural process with $A_{0}=0$.

Proof. This is a combination of Theorems \ref{prot35} and \ref{prot36}. $\blacksquare$

\begin{equation}{\label{prot37}}\tag{3}\mbox{}\end{equation}

Theorem \ref{prot37}. (The Doob-Meyer Decomposition). Let $Z$ be a supermartingale. Then $Z$ has a decomposition $Z=Z_{0}+M-A$ where $M$ is a local martingale and $A$ is an increasing process which is locally natural, and $M_{0}=A_{0}=0$. Such a decomposition is unique. Moreover if $\lim_{t\rightarrow\infty}E[Z_{t}]>-\infty$, then $\mathbb{E}[A_{\infty}]<\infty$, and $A$ is natural. $\sharp$

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

Quasimartingales.

Let $X$ be a RCLL, adapted process defined on $[0,\infty ]$. It is convenient when discussing quasimartingales to include $\infty$ in the index set, thue making it homeomorphic to $[0,t]$ for $0<t\leq\infty$. If a process $X$ is defined only on $[0,\infty )$ we extend it to $[0,\infty ]$ by setting $X_{\infty}=0$.

Definition. A finite tuple of points $\pi =\{t_{0},t_{1},\cdots ,t_{n+1}\}$ satisfying $0=t_{0}<t_{1}<\cdots <t_{n+1}=\infty$ is a {\bf partition} of $[0,\infty ]$. $\sharp$

Definition. Suppose that $\pi$ is a partition of $[0,\infty ]$ and that $X_{t_{i}}\in L^{1}$ for each $t_{i}\in\pi$. Define
\[C(X,\pi )=\sum_{i=0}^{n}\left |E\left [\left .X_{t_{i}}-X_{t_{i+1}}\right |{\cal F}_{t_{i}}\right ]\right |;\]
the {\bf variation of $X$ along} $\pi$ is defined to be
\[V_{\pi}[X]=E[C(X,\pi )].\]
The {\bf variation of $X$} is defined to be
\[V(X)=\sup_{\pi}V_{\pi}(X),\]
where the supremum is taken over all such partitions. $\sharp$

Definition. An adapted, RCLL process $X$ is a quasimartingale on $[0,\infty )$ if $\mathbb{E}|X_{t}|]<\infty$ for each $t$, and if $\mathbb{V}(X)<\infty$. $\sharp$

Recall that by convention if $X$ is defined only on $[0,\infty )$, we set $X_{\infty}=0$.

\begin{equation}{\label{prot38}}\tag{4}\mbox{}\end{equation}

Proposition \ref{prot38}. Let $X$ be a process indexed by $[0,\infty )$. Then $X$ is a quasimartingale if and only if $X$ has a decomposition $X=Y-Z$ where $Y$ and $Z$ are each positive right-continuous supermartingale. $\sharp$

\begin{equation}{\label{prot39}}\tag{5}\mbox{}\end{equation}

Theorem \ref{prot39}. A quasimartingale $X$ has a unique decomposition $X=M+A$ where $M$ is a local martingale and $A$ is a locally natural process with paths of finite variation on compacts and $A_{0}=0$.

Proof. This result is a combination of Theorem \ref{prot37} and Proposition \ref{prot38}. $\blacksquare$

If $A$ is of locally integrable variation it is then a locally quasimartingale, and hence by Rao’s theorem (Theorem \ref{prot39}), there exists a unique dcomposition $A=M+\tilde{A}$, where $\tilde{A}$ is a
locally natural FV process. That is, there exists a unique, locally natural (predictable) FV process $\tilde{A}$ such that $A-\tilde{A}$ is a local martingale.

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

The Fundamental Theorem of Local Martingales.

Definition. A stopping time $T$ is {\bf predictable} if there exists a sequence of stopping times $\{T_{n}\}_{n\in {\bf N}}$ such that $T_{n}$ is increasing, $T_{n}<T$ on $\{T>0\}$ for all $n$, and $\lim_{n\rightarrow\infty}T_{n}=T$ a.s. such a sequence $\{T_{n}\}_{n\in {\bf N}}$ is said to {\bf announce} $T$. $\sharp$

If $X$ is a continuous, adapted process with $X_{0}=0$, and $T=\inf\{t:|X_{t}|\geq c\}$ for some $c>0$, then $T$ is predictable; indeed, the sequence
\[T_{n}=\inf\left\{t:|X_{t}|\geq c-\frac{1}{n}\right\}\wedge n\]
is an announcing sequence. Fixed times are also predictable.

Definition. A stopping time $T$ is {\bf accessible} if there exists a sequence $\{T_{n}\}_{n\in {\bf N}}$ of predictable times such that
\[P\left\{\bigcup_{n=1}^{\infty}\{\omega :T_{n}(\omega )=T(\omega )<\infty\}\right\}=P\{T<\infty\}.\]
Such a sequence $\{T_{n}\}_{n\in {\bf N}}$ is said to envelop $T$. $\sharp$

Any stopping time that takes on a countable number of values is clearly accessible.

Definition. A stopping time $T$ is {\bf totally inaccessible} if for every predictable stopping time $S$,
\[P\{\omega :T(\omega )=S(\omega )<\infty\}=0. \sharp\]

Let $T$ be a stopping time and $\Gamma\in {\cal F}_{T}$. We define
\[T_{\Gamma}(\omega )=\left\{\begin{array}{ll}
T(\omega ) & \mbox{if $\omega\in\Gamma$}\\
\infty & \mbox{if $\omega\not\in\Gamma$}
\end{array}\right .\]
It is simple to check that since $\Gamma\in {\cal F}_{T}$, $T_{\Gamma}$ is a stopping time. Note further that $T=\min\{T_{\Gamma},T_{\Gamma^{c}}\}$.

Proposition. Let $T$ be a stopping time. There exist disjoint events $A$ and $B$ such that $A\cup B=\{T<\infty\}$ a.s., such that $T_{A}$ is accessible and $T_{B}$ is totally inaccessible; and $T=T_{A}\wedge T_{B}$ a.s. Such a decomposition is unique a.s. $\sharp$

Theorem. (Fundamental Theorem of Local Martingales). Let $M$ be a local martingale and let $\beta >0$. Then there exist local martingales $N$ and $A$ such that $A$ is an FV process, the jumps of $N$ are bounded by $2\beta$, and $M=N+A$. $\sharp$

Corollary. A local martingale is decomposable. $\sharp$

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

Classical Semimartingales.

We have seen that a decomposable process is a semimartingale.

\begin{equation}{\label{prot314}}\tag{6}\mbox{}\end{equation}

Proposition \ref{prot314}. A classical semimartingale is a semimartingale. $\sharp$

\begin{equation}{\label{proc314}}\tag{7}\mbox{}\end{equation}

Corollary \ref{proc314}. A RCLL local martingale is a semimartingale.

Proof. A local martingale is a classical semimartingale. $\blacksquare$

\begin{equation}{\label{prot315}}\tag{8}\mbox{}\end{equation}

Proposition \ref{prot315}. A RCLL quasimartingale is semimartingale.

Proof. By Theorem \ref{prot39}, a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Proposition \ref{prot314}. $\blacksquare$

Proposition. A RCLL supermartingale is a semimartingale.

Proof. Since a local martingale is a semimartingale, it suffices to show that for a supermatingale $X$, the stopped process $X^{t}$ is a semimartingale. However for a partition $\pi$ of $[0,t]$,
\begin{align*}
\mathbb{E}\left [\sum_{t_{i}\in\pi}|\mathbb{E}[X_{t_{i}}-X_{t_{i+1}}|{\cal F}_{t_{i}}]|\right ] & =\mathbb{E}\left [\sum_{t_{i}\in\pi}\mathbb{E}[X_{t_{i}}-X_{t_{i+1}}|{\cal F}_{t_{i}}] \right ]\\
& =\sum_{t_{i}\in\pi}\left (\mathbb{E}[X_{t_{i}}]-\mathbb{E}[X_{t_{i+1}}]\right )=\mathbb{E}[X_{0}]-\mathbb{E}[X_{t}].
\end{align*}
Therefore $X^{t}$ is a quasimartingale, hence s semimartingale by Proposition \ref{prot315}. $\blacksquare$

Corollary. A submartingale is a semimartingale. $\sharp$

Proposition. Let $M$ be a local martingale and let $H\in {\bf L}$. Then the stochastic integral $H\bullet M$ is again a local martingale. $\sharp$

Let $X$ be a classical semimartingale, and let $X_{t}=X_{0}+M_{t}+A_{t}$ be a decomposition where $M_{0}=A_{0}=0$, $M$ is a local martingale, and $A$ is an FV process. We see that the decomposition of a classical semimartingale need not be unique. This problem can often be solved by choosing a certain canonical decomposition which is unique.

Definition. Let $X$ be a semimartingale. If $X$ has a decomposition $X_{t}=X_{0}+M_{t}+A_{t}$ with $M_{0}=A_{0}=0$, $M$ a local martingale, $A$ an FV process, and with $A$ locally natural, then $X$ is said to be a special semimartingale. $\sharp$

We assume $X_{0}=0$.

\begin{equation}{\label{prot318}}\tag{9}\mbox{}\end{equation}

Proposition \ref{prot318}. If $X$ is a special semimartingale, then its decomposition $X=M+A$ with $A$ locally natural is unique. $\sharp$

Definition. If $X$ is a special semimartingale, then the unique decomposition $X=M+A$ with $A$ locally natural is called the canonical decomposition. $\sharp$

Theorem \ref{prot39} shows that any quasimartingale is special. A useful sufficient condition for a semimartingale $X$ to be special is that $X$ be a classical semimartingale, or equivalently decomposable, and also have bounded jumps.

Proposition. Let $X$ be a classical semimartingale with bounded jumps. Then $X$ is a special semimartingale. $\sharp$

Corollary. Let $X$ be a classical semimartingale with continuous paths. Then $X$ is special and in its canonical decomposition
\[X_{t}=X_{0}+M_{t}+A_{t},\]
the local martingale $M$ and the FV process $A$ have continuous paths. $\sharp$

We saw that a classical semimartingale is a semimartingale in Proposition \ref{prot314}. Now we have the converse.

\begin{equation}{\label{prot322}}\tag{10}\mbox{}\end{equation}

Theorem \ref{prot322}. (Bichteler-Dellacherie). An Adapted, RCLL process $X$ is a semimartingale if and only if it is a classical semimartingale. That is, $X$ is a semimartingale if and only if it can be written $X=M+A$, where $M$ is a local martingale and $A$ is an FV process. $\sharp$

We state again, for emphasis, that Proposition \ref{prot314} and Theorem \ref{prot322} together allows us to conclude that semimartingales and classical semimartingales are the same.

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

Natural Versus Predictable Processes.

Recall that ${\bf L}$ is the space of adapted processes having left-continuous paths with right limits

Definition. The predictable $\sigma$-algebra ${\cal P}$ on $\mathbb{R}_{+}\times\Omega$ is the smallest $\sigma$-algebra making all processes in ${\bf L}$ measurable. We also let $({\bf b}{\cal P})$ ${\cal P}$ denote the (bounded) processes that are predictably measurable. $\sharp$

Proposition. Let $A$ be an FV process of integrable variation with $A_{0}=0$. If $A$ is predictable, then $A$ is natural. $\sharp$

Corollary. Let $A$ be an FV process of locally integrable variation with $A_{0}=0$. If $A$ is predictable, then $A$ is natural.

Corollary. Let $A$ be an FV process of locally integrable variation with $A_{0}=0$. If $A$ is predictable and a local martingale, then $A$ is identically zero. $\sharp$

Definition. Let $T$ be a stopping time. The $\sigma$-filed ${\cal F}_{T-}$ is the smallest $\sigma$-filed containing ${\cal F}_{0}$ and all sets of the form $A\cap\{t<T\}$ for $t>0$ and $A\in {\cal F}_{t}$. Observe that ${\cal F}_{T-}\subset {\cal F}_{T}$, and also the stopping times $T$ is ${\cal F}_{T-}$-measurable. $\sharp$

Proposition. Let $H$ be a bounded and measurable process. There exists a unique predictable process $\bar{H}$ bounded by $\sup|H|$ such that
\[\bar{H}_{T}=E[H_{T}|{\cal F}_{T-}]\mbox{ on }\{T<\infty\}\]
for all predictable stopping times $T$. $\sharp$

Definition. Let $H$ be a bounded and measurable process. The predictable projection of $H$, written $\bar{H}$, is the unique predictable process such that
\[\bar{H}_{T}=E[H_{t}|{\cal F}_{T-}]\mbox{ on }\{T<\infty\}\]
for all predictable stopping times $T$. $\sharp$

Proposition. Let $V$ be a natural process $($of integrable variation$)$, and let $H$ be bounded and measurable. Then
\[\mathbb{E}\left [\int_{0}^{\infty}H_{s}dV_{s}\right ]=\mathbb{E}\left [\int_{0}^{\infty}\bar{H}_{s}dV_{s}\right ],\]
where the stochastic integral are taken in the Stieltjes snse, path by path. $\sharp$

Proposition. Let $A$ be a bounded natural process. Then $A$ is predictable. $\sharp$

Proposition. Let $A$ be a bounded FV process of locally integrable variation with $A_{0}=0$. Then $A$ is predictable if and only if $A$ is locally natural. $\sharp$