Almost Sure

22 December 09

Class (D) Processes

A stochastic process X is said to be uniformly integrable if the set of random variables {\{X_t\colon t\in{\mathbb R}_+\}} is uniformly integrable. However, even if this is the case, it does not follow that the set of values of the process sampled at arbitrary stopping times is uniformly integrable.

For the case of a cadlag martingale X, optional sampling can be used. If {t\ge 0} is any fixed time then this says that {X_\tau={\mathbb E}[X_t\mid\mathcal{F}_\tau]} for stopping times {\tau\le t}. As sets of conditional expectations of a random variable are uniformly integrable, the following result holds.

Lemma 1 Let X be a cadlag martingale. Then, for each {t\ge 0}, the set

\displaystyle  \{X_\tau\colon\tau\le t\text{\ is\ a\ stopping\ time}\}

is uniformly integrable.

This suggests the following generalized concepts of uniform integrability for stochastic processes.

Definition 2 Let X be a jointly measurable stochastic process. Then, it is

  • of class (D) if {\{X_\tau\colon\tau<\infty\text{ is a stopping time}\}} is uniformly integrable.
  • of class (DL) if, for each {t\ge 0}, {\{X_\tau\colon\tau\le t\text{ is a stopping time}\}} is uniformly integrable.

The term `class (D)’ is used extensively in the literature on stochastic process theory and, in particular, class (D) submartingales are commonly used for the Doob-Meyer decomposition. The origin of the term is not clear, although it has been suggested that Meyer originally used the term in reference to Doob, concerning processes for which Doob’s decomposition results can be generalized. The term `class (DL)’ is then a kind of localization of class (D). A process X is of class (DL) if and only if the stopped processes {X^t} are of class (D) for all times {t}.

For martingales, the following is true.

Lemma 3 Any cadlag martingale X is of class (DL).

Furthermore, X is of class (D) if and only if it is uniformly integrable.

This result follows from Lemma 4 below applied to the nonnegative submartingale {\vert X\vert}.

Before proceeding, I mention the following basic properties of uniform integrability, which are used in this post. This is just standard measure theory, and I added proofs of these to the PlanetMath website.

  1. The set {\{Z\}} is uniformly integrable for any given integrable variable Z.
  2. Any collection of random variables Y which are all dominated by random variables {Z\in S} from some uniformly integrable set {S} (so that {\vert Y\vert\le\vert Z\vert}) is itself uniformly integrable.
  3. Any collection of conditional expectations of random variables in a uniformly integrable set {S} is itself uniformly integrable.
  4. The set of limits (in probability) of random variables in a uniformly integrable set {S} is itself uniformly integrable.

As for martingales, Nonnegative submartingales satisfy particularly good uniform integrability properties.

Lemma 4 Any positive cadlag submartingale X is of class (DL).

Furthermore, X is of class (D) if and only if it is uniformly integrable.

Proof: By optional sampling, for any stopping {\tau\le t}, the inequality

\displaystyle  0\le X_\tau\le{\mathbb E}[X_t\mid\mathcal{F}_\tau]

holds. So, the set of random variables {X_\tau} for such stopping times are dominated by conditional expectations of {X_t}, hence is uniformly integrable. Therefore the process is of class (DL).

Now suppose that {\{X_t\colon t\in{\mathbb R}_+\}} is uniformly integrable. By the argument above, the set of random variables {X_\tau} for bounded stopping times are dominated by conditional expectations of this set. Furthermore, for finite stopping times, {X_\tau=\lim_{n\rightarrow\infty}X_{\tau\wedge n}} is a limit of random variables from this set and X is of class (D). Conversely, if X is of class (D) then sampling X at the constant stopping times shows that {\{X_t\colon t\in{\mathbb R}_+\}} is uniformly integrable. \Box

The conclusion of Lemma 4 does not hold for arbitrary cadlag submartingales, so the nonnegativity condition is required. For counterexamples, see the examples of local martingales which are not proper martingales from the following post (they are submartingales, and not of class (DL)). However, the following is true.

Lemma 5 A cadlag submartingale is of class (DL) if and only if its negative part is of class (DL).

Proof: If X is a cadlag submartingale then its positive part {X^+} is a nonnegative submartingale which, by Lemma 4, is of class (DL). Writing {X=X^+-X^-} shows that X is class (DL) if and only if {X^-} is. \Box

Given an {L^p}-integrable martingale X, for {p>1}, it follows that {\vert X \vert^p} is a nonnegative submartingale and, hence, of class (DL). In fact, by Doob’s martingale inequalities, we have the much stronger condition that {\vert X\vert^p} is dominated in {L^1} by {\sup_{s\le t}\vert X_s\vert}. In any case, the following simple corollary is useful.

Corollary 6 If X is an {L^p}-integrable cadlag martingale then {|X|^p} is of class (DL).

For arbitrary cadlag submartingales, where the class (DL) property can fail, the following result concerning uniform integrability at a decreasing sequence of stopping times can sometimes be applied instead.

Lemma 7 Let X be a cadlag submartingale and {\tau_n} be a decreasing sequence of bounded stopping times. Then, {\{X_{\tau_n}\colon n=1,2,\ldots\}} is uniformly integrable.

Proof: The following result can be applied; a submartingale sampled at a decreasing sequence of times which are bounded below is a uniformly integrable sequence. This was proven in a previous post using a simple `Doob-style’ decomposition.

In this case, for each positive integer n set {Y_{-n}=X_{\tau_n}}. By optional sampling, is a submartingale with time index running over the negative integers, and with respect to the filtration {\mathcal{G}_{-n}=\mathcal{F}_{\tau_n}}. Setting {\tau_\infty\equiv 0} extends {Y_t} to {t=-\infty}, bounding the negative integers from below. So, {X_{\tau_n}=Y_{-n}} is a uniformly integrable sequence. \Box

4 Comments »

  1. Hi,

    A little remark, you mention a counter example in some post after the proof of lemma 4, but the hyperlink is missing.

    Regards

    Comment by TheBridge — 19 October 11 @ 1:52 PM | Reply

    • When I said `the following post’ I meant to say `the upcoming post in these notes’. So, it was not posted until after this one and couldn’t be linked when this was posted as it didn’t exist yet. I linked it now. Thanks for pointing this out.

      Comment by George Lowther — 24 October 11 @ 3:06 AM | Reply

  2. X^2 is a submartingale if X is a martingale? How does that follow? surely we only know it is a local submartingale ?

    Comment by yufan — 30 October 12 @ 8:58 PM | Reply

    • I’m probably missing something but doesn’t this just follow by Jensen?

      Comment by idontgetoutmuch — 2 November 12 @ 8:55 AM | Reply


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