Chaotic and predictable representations for Lévy processes

Abstract

The only normal martingales which posses the chaotic representation property and the weaker predictable representation property and which are at the same time also Lévy processes, are in essence Brownian motion and the compensated Poisson process. For a general Lévy process (satisfying some moment conditions), we introduce the power jump processes and the related Teugels martingales. Furthermore, we orthogonalize the Teugels martingales and show how their orthogonalization is intrinsically related with classical orthogonal polynomials. We give a chaotic representation for every square integral random variable in terms of these orthogonalized Teugels martingales. The predictable representation with respect to the same set of orthogonalized martingales of square integrable random variables and of square integrable martingales is an easy consequence of the chaotic representation.