The Explicit Chaotic Representation of the powers of increments of Levy Processes

An explicit formula for the chaotic representation of the powers of increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are two different chaos expansions of a square integrable functional of a Levy process: one with respect to the compensated Poisson random measure and the other with respect to the orthogonal compensated powers of the jumps of the Levy process. Computationally explicit formulae for both of these chaos expansions of (X_{t+t_0}-X_{t_0})^n are given in this paper. Simulation results verify that the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of (X_{t+t_0}-X_{t_0})^n using Taylor's expansion.