White Noise of Poisson Random Measures

Abstract

We develop a white noise theory for Poisson random measures associated with a Lévy process. The starting point of this theory is a chaos expansion with kernels of polynomial type. We use this to construct the white noise of a Poisson random measure, which takes values in a certain distribution space. Then we show, how a Skorohod/Itô integral for point processes can be represented by a Bochner integral in terms of white noise of the random measure and a Wick product. Further, we apply these concepts to derive a generalized Clark-Haussmann-Ocone theorem for Lévy processes. Finally, an application of this theorem to portfolios in financial markets, driven by Lévy processes, is given.

Key words and phrases: Lévy processes, white noise analysis, stochastic partial differential equations