White noise analysis for Levy processes

摘要

We construct a white noise theory for Levy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula for Levy processes F(omega) = E[F] +Sigmamgreater than or equal to1 integral(0)(T) E[D-t((m)) FF-t] lozenge Y-t((m)) dt. Here E[F] is the generalized expectation, the operators Dt(m)F, mgreater than or equal to1 are (generalized) Malliavin derivatives, lozenge is the Wick product and for all mgreater than or equal to1 Y-t((m)) is the white noise of power jump processes Y-t((m)). In particular, Y-t((1)) is the white noise of the Levy process. The formula holds for all F is an element of G* superset of L-2(mu), where G* is a space of stochastic distributions and mu is a white noise probability measure. Finally, we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by Levy processes. (C) 2003 Elsevier Inc. All rights reserved. [References: 35]