The Chaotic Representation Property of Compensated-Covariation Stable Families of Martingales

TL;DR

This paper establishes conditions under which families of square integrable martingales have the chaotic representation property, with applications to Lévy processes, Gaussian martingales, and Poisson processes.

Contribution

It introduces the notion of compensated-covariation stability and proves the chaotic representation property for such families under deterministic covariation conditions.

Findings

Families with compensated-covariation stability have the chaotic representation property.

Application of the main result to Lévy processes and construction of martingale families.

Examples include Teugels martingales and Gaussian families.

Abstract

In the present paper, we study the chaotic representation property for certain families of square integrable martingales. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family of square integrable martingales having deterministic mutual predictable covariation. The main result of the present paper is: If $\mathscr{X}$ is a compensated-covariation stable family of square integrable martingales such that $\langle{X},{Y}\rangle$ is deterministic for all $X,Y\in\mathscr{X}$ and, furthermore, the system of monomials generated by $\mathscr{X}$ is total in $L^2(\Omega,\mathscr{F}^\mathscr{X}_T,\mathbb{P})$, then $\mathscr{X}$ possesses the chaotic representation property. We shall apply this result to the case of L\'evy processes. Relative to the filtration generated by a L\'evy process, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of L\'evy processes, several examples of concrete families $\mathscr{X}$ of martingales including Teugels martingales.