The chaotic representation property of compensated-covariation stable families of martingales

Abstract

In the present paper, we study the chaotic representation property for certain families ${\mathscr{X}}$ of square integrable martingales on a finite time interval $[0,T]$. 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 ${\mathscr{X}}$ of square integrable martingales having deterministic mutual predictable covariation $\langle X,Y\rangle$ for all $X,Y\in{\mathscr{X}}$. The main result of the present paper is stated in Theorem 5.8 below: 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 with respect to the $\sigma$-field $\mathscr{F}^{\mathscr{X}}_{T}$. We shall apply this result to the case of Lévy processes. Relative to the filtration $\mathbb{F}^{L}$ generated by a Lévy process $L$, 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 Lévy processes, several examples of concrete families ${\mathscr{X}}$ of martingales including Teugels martingales.