The Doob-Meyer Decomposition

The Doob-Meyer decomposition was a very important result, historically, in the development of stochastic calculus. This theorem states that every cadlag submartingale uniquely decomposes as the sum of a local martingale and an increasing predictable process. For one thing, if X is a square-integrable martingale then Jensen’s inequality implies that ${X^2}$ is a submartingale, so the Doob-Meyer decomposition guarantees the existence of an increasing predictable process ${\langle X\rangle}$ such that ${X^2-\langle X\rangle}$ is a local martingale. The term ${\langle X\rangle}$ is called the predictable quadratic variation of X and, by using a version of the Ito isometry, can be used to define stochastic integration with respect to square-integrable martingales. For another, semimartingales were historically defined as sums of local martingales and finite variation processes, so the Doob-Meyer decomposition ensures that all local submartingales are also semimartingales. Going further, the Doob-Meyer decomposition is used as an important ingredient in many proofs of the Bichteler-Dellacherie theorem.

The approach taken in these notes is somewhat different from the historical development, however. We introduced stochastic integration and semimartingales early on, without requiring much prior knowledge of the general theory of stochastic processes. We have also developed the theory of semimartingales, such as proving the Bichteler-Dellacherie theorem, using a stochastic integration based method. So, the Doob-Meyer decomposition does not play such a pivotal role in these notes as in some other approaches to stochastic calculus. In fact, the special semimartingale decomposition already states a form of the Doob-Meyer decomposition in a more general setting. So, the main part of the proof given in this post will be to show that all local submartingales are semimartingales, allowing the decomposition for special semimartingales to be applied.

The Doob-Meyer decomposition is especially easy to understand in discrete time, where it reduces to the much simpler Doob decomposition. If ${\{X_n\}_{n=0,1,2,\ldots}}$ is an integrable discrete-time process adapted to a filtration ${\{\mathcal{F}_n\}_{n=0,1,2,\ldots}}$ , then the Doob decomposition expresses X as

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle X_n&\displaystyle=M_n+A_n,\smallskip\\ \displaystyle A_n&\displaystyle=\sum_{k=1}^n{\mathbb E}\left[X_k-X_{k-1}\;\vert\mathcal{F}_{k-1}\right]. \end{array}$

(1)

As previously discussed, M is then a martingale and A is an integrable process which is also predictable, in the sense that ${A_n}$ is ${\mathcal{F}_{n-1}}$ -measurable for each ${n > 0}$ . Furthermore, X is a submartingale if and only if ${{\mathbb E}[X_n-X_{n-1}\vert\mathcal{F}_{n-1}]\ge0}$ or, equivalently, if A is almost surely increasing.

Moving to continuous time, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$ with time index t ranging over the nonnegative real numbers. Then, the continuous-time version of (1) takes A to be a right-continuous and increasing process which is predictable, in the sense that it is measurable with respect to the σ-algebra generated by the class of left-continuous and adapted processes. Often, the Doob-Meyer decomposition is stated under additional assumptions, such as X being of class (D) or satisfying some similar uniform integrability property. To be as general possible, the statement I give here only requires X to be a local submartingale, and furthermore states how the decomposition is affected by various stronger hypotheses that X may satisfy.

Theorem 1 (Doob-Meyer) Any local submartingale X has a unique decomposition

$\displaystyle X=M+A,$ (2)

where M is a local martingale and A is a predictable increasing process starting from zero.

Furthermore,

if X is a proper submartingale, then A is integrable and satisfies

$\displaystyle {\mathbb E}[A_\tau]\le{\mathbb E}[X_\tau-X_0]$ (3)

for all uniformly bounded stopping times ${\tau}$ .

X is of class (DL) if and only if M is a proper martingale and A is integrable, in which case

$\displaystyle {\mathbb E}[A_\tau]={\mathbb E}[X_\tau-X_0]$ (4)

for all uniformly bounded stopping times ${\tau}$ .

X is of class (D) if and only if M is a uniformly integrable martingale and ${A_\infty}$ is integrable. Then, ${X_\infty=\lim_{t\rightarrow\infty}X_t}$ and ${M_\infty=\lim_{t\rightarrow\infty}M_t}$ exist almost surely, and (4) holds for all (not necessarily finite) stopping times ${\tau}$ .

Continue reading “The Doob-Meyer Decomposition” →

Compensators of Counting Processes

A counting process, X, is defined to be an adapted stochastic process starting from zero which is piecewise constant and right-continuous with jumps of size 1. That is, letting ${\tau_n}$ be the first time at which ${X_t=n}$ , then

$\displaystyle X_t=\sum_{n=1}^\infty 1_{\{\tau_n\le t\}}.$

By the debut theorem, ${\tau_n}$ are stopping times. So, X is an increasing integer valued process counting the arrivals of the stopping times ${\tau_n}$ . A basic example of a counting process is the Poisson process, for which ${X_t-X_s}$ has a Poisson distribution independently of ${\mathcal{F}_s}$ , for all times ${t > s}$ , and for which the gaps ${\tau_n-\tau_{n-1}}$ between the stopping times are independent exponentially distributed random variables. As we will see, although Poisson processes are just one specific example, every quasi-left-continuous counting process can actually be reduced to the case of a Poisson process by a time change. As always, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$ .

Note that, as a counting process X has jumps bounded by 1, it is locally integrable and, hence, the compensator A of X exists. This is the unique right-continuous predictable and increasing process with ${A_0=0}$ such that ${X-A}$ is a local martingale. For example, if X is a Poisson process of rate ${\lambda}$ , then the compensated Poisson process ${X_t-\lambda t}$ is a martingale. So, the compensator of X is the continuous process ${A_t=\lambda t}$ . More generally, X is said to be quasi-left-continuous if ${{\mathbb P}(\Delta X_\tau=0)=1}$ for all predictable stopping times ${\tau}$ , which is equivalent to the compensator of X being almost surely continuous. Another simple example of a counting process is ${X=1_{[\tau,\infty)}}$ for a stopping time ${\tau > 0}$ , in which case the compensator of X is just the same thing as the compensator of ${\tau}$ .

As I will show in this post, compensators of quasi-left-continuous counting processes have many parallels with the quadratic variation of continuous local martingales. For example, Lévy’s characterization states that a local martingale X starting from zero is standard Brownian motion if and only if its quadratic variation is ${[X]_t=t}$ . Similarly, as we show below, a counting process is a homogeneous Poisson process of rate ${\lambda}$ if and only if its compensator is ${A_t=\lambda t}$ . It was also shown previously in these notes that a continuous local martingale X has a finite limit ${X_\infty=\lim_{t\rightarrow\infty}X_t}$ if and only if ${[X]_\infty}$ is finite. Similarly, a counting process X has finite value ${X_\infty}$ at infinity if and only if the same is true of its compensator. Another property of a continuous local martingale X is that it is constant over all intervals on which its quadratic variation is constant. Similarly, a counting process X is constant over any interval on which its compensator is constant. Finally, it is known that every continuous local martingale is simply a continuous time change of standard Brownian motion. In the main result of this post (Theorem 5), we show that a similar statement holds for counting processes. That is, every quasi-left-continuous counting process is a continuous time change of a Poisson process of rate 1. Continue reading “Compensators of Counting Processes” →

Compensators of Stopping Times

The previous post introduced the concept of the compensator of a process, which is known to exist for all locally integrable semimartingales. In this post, I’ll just look at the very special case of compensators of processes consisting of a single jump of unit size.

Definition 1 Let ${\tau}$ be a stopping time. The compensator of ${\tau}$ is defined to be the compensator of ${1_{[\tau,\infty)}}$ .

So, the compensator A of ${\tau}$ is the unique predictable FV process such that ${A_0=0}$ and ${1_{[\tau,\infty)}-A}$ is a local martingale. Compensators of stopping times are sufficiently special that we can give an accurate description of how they behave. For example, if ${\tau}$ is predictable, then its compensator is just ${1_{\{\tau > 0\}}1_{[\tau,\infty)}}$ . If, on the other hand, ${\tau}$ is totally inaccessible and almost surely finite then, as we will see below, its compensator, A, continuously increases to a value ${A_\infty}$ which has the exponential distribution.

However, compensators of stopping times are sufficiently general to be able to describe the compensator of any cadlag adapted process X with locally integrable variation. We can break X down into a continuous part plus a sum over its jumps,

$\displaystyle X_t=X_0+X^c_t+\sum_{n=1}^\infty\Delta X_{\tau_n}1_{[\tau_n,\infty)}.$

(1)

Here, ${\tau_n > 0}$ are disjoint stopping times such that the union ${\bigcup_n[\tau_n]}$ of their graphs contains all the jump times of X. That they are disjoint just means that ${\tau_m\not=\tau_n}$ whenever ${\tau_n < \infty}$ , for any ${m\not=n}$ . As was shown in an earlier post, not only is such a sequence ${\tau_n}$ of the stopping times guaranteed to exist, but each of the times can be chosen to be either predictable or totally inaccessible. As the first term, ${X^c_t}$ , on the right hand side of (1) is a continuous FV process, it is by definition equal to its own compensator. So, the compensator of X is equal to ${X^c}$ plus the sum of the compensators of ${\Delta X_{\tau_n}1_{[\tau_n,\infty)}}$ . The reduces compensators of locally integrable FV processes to those of processes consisting of a single jump at either a predictable or a totally inaccessible time. Continue reading “Compensators of Stopping Times” →

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