  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.
