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In the present paper, we study the chaotic representation property forcertain families of square integrable martingales. For this purpose, weintroduce the notion of compensated-covariation stability of such families. Thechaotic representation property will be defined using iterated integrals withrespect to a given family of square integrable martingales having deterministicmutual predictable covariation. The main result of the present paper is: If$\mathscr{X}$ is a compensated-covariation stable family of square integrablemartingales 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 shallapply this result to the case of L\'evy processes. Relative to the filtrationgenerated by a L\'evy process, we construct families of martingales whichpossess the chaotic representation property. As an illustration of the generalresults, we will also discuss applications to continuous Gaussian families ofmartingales and independent families of compensated Poisson processes. Weconclude the paper by giving, for the case of L\'evy processes, severalexamples of concrete families $\mathscr{X}$ of martingales including Teugelsmartingales.

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