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 XX of square integrable martingales on a finite time interval [0,T][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 XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, 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 Levy processes, several examples of concrete families XX of martingales including Teugels martingales.