Month: April 2012

Quasimartingales

Quasimartingales are a natural generalization of martingales, submartingales and supermartingales. They were first introduced by Fisk in order to extend the Doob-Meyer decomposition to a larger class of processes, showing that continuous quasimartingales can be decomposed into martingale and finite variation terms (Quasi-martingales, 1965). This was later extended to right-continuous processes by Orey (F-Processes, 1967). The way in which quasimartingales relate to sub- and super-martingales is very similar to how functions of finite variation relate to increasing and decreasing functions. In particular, by the Jordan decomposition, any finite variation function on an interval decomposes as the sum of an increasing and a decreasing function. Similarly, a stochastic process is a quasimartingale if and only if it can be written as the sum of a submartingale and a supermartingale. This important result was first shown by Rao (Quasi-martingales, 1969), and means that much of the theory of submartingales can be extended without much work to also cover quasimartingales.

Often, given a process, it is important to show that it is a semimartingale so that the techniques of stochastic calculus can be applied. If there is no obvious decomposition into local martingale and finite variation terms, then, one way of doing this is to show that it is a quasimartingale. All right-continuous quasimartingales are semimartingales. This result is also important in the general theory of semimartingales with, for example, many proofs of the Bichteler-Dellacherie theorem involving quasimartingales.

In this post, I will mainly be concerned with the definition and very basic properties of quasimartingales, and look at the more advanced theory in the following post. We work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}. It is not necessary to assume that either of the usual conditions, of right-continuity or completeness, hold. First, the mean variation of a process is defined as follows.

Definition 1 The mean variation of an integrable stochastic process X on an interval {[0,t]} is

\displaystyle {\rm Var}_t(X)=\sup{\mathbb E}\left[\sum_{k=1}^n\left\vert{\mathbb E}\left[X_{t_k}-X_{t_{k-1}}\;\vert\mathcal{F}_{t_{k-1}}\right]\right\vert\right]. (1)

Here, the supremum is taken over all finite sequences of times,

\displaystyle 0=t_0\le t_1\le\cdots\le t_n=t.

A quasimartingale, then, is a process with finite mean variation on each bounded interval.

Definition 2 A quasimartingale, X, is an integrable adapted process such that {{\rm Var}_t(X)} is finite for each time {t\in{\mathbb R}_+}.

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