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

Abstract

We study the predictable representation property for certain families of square integrable martingales, which we call compensated-covariation stable families. The definition of the predictable representation property is given by means of stable subspaces. The main result is that compensated-covariation stable families of martingales which satisfy some further conditions possess the predictable representation property. As first examples, we consider continuous Gaussian families of martingales and independent families of compensated Poisson processes. Then we apply the result to the case of Lévy processes. We shall construct families of martingales relative to a Lévy filtration which possess the predictable representation property. We give several examples including Teugels martingales.