Almost Sure

18 July 16

The Doob-Meyer Decomposition for Quasimartingales

As previously discussed, for discrete-time processes the Doob decomposition is a simple, but very useful, technique which allows us to decompose any integrable process into the sum of a martingale and a predictable process. 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)

Then, 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}. The expected value of the variation of A can be computed in terms of X,

\displaystyle  {\mathbb E}\left[\sum_{k=1}^n\lvert A_k-A_{k-1}\rvert\right] ={\mathbb E}\left[\sum_{k=1}^n\left\lvert {\mathbb E}[X_k-X_{k-1}\vert\;\mathcal{F}_{k-1}]\right\rvert\right].

This is the mean variation of X.

In continuous time, the situation is rather more complex, and will require constraints on the process X other than just integrability. We have already discussed the case for submartingales — the Doob-Meyer decomposition. This decomposes a submartingale into a local martingale and a predictable increasing process.

A natural setting for further generalising the Doob-Meyer decomposition is that of quasimartingales. In continuous time, the appropriate class of processes to use for the component A of the decomposition is the predictable FV processes. Decomposition (2) below is the same as that in the previous post on special semimartingales. This is not surprising, as we have already seen that the class of special semimartingales is identical to the class of local quasimartingales. The difference with the current setting is that we can express the expected variation of A in terms of the mean variation of X, and obtain a necessary and sufficient condition for the local martingale component to be a proper martingale.

As was noted in an earlier post, historically, decomposition (2) for quasimartingales played an important part in the development of stochastic calculus and, in particular, in the proof of the Bichteler-Dellacherie theorem. That is not the case in these notes, however, as we have already proven the main results without requiring quasimartingales. As always, any two processes are identified whenever they are equivalent up to evanescence.

Theorem 1 Every cadlag quasimartingale X uniquely decomposes as

\displaystyle  X=M+A (2)

where M is a local martingale and A is a predictable FV process with {A_0=0}. Then, A has integrable variation over each finite time interval {[0,t]} satisfying

\displaystyle  {\rm Var}_t(X)={\rm Var}_t(M)+{\mathbb E}\left[\int_0^t\,\vert dA\vert\right]. (3)

so that, in particular,

\displaystyle  {\mathbb E}\left[\int_0^t\,\vert dA\vert\right]\le{\rm Var}_t(X). (4)

Furthermore, the following are equivalent,

  1. X is of class (DL).
  2. M is a proper martingale.
  3. inequality (4) is an equality for all times t.

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30 December 11

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,

  1. 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}.

  2. 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}.

  3. 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}.

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