White noise analysis for Lévy processes

Abstract

We construct a white noise theory for Lévy 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 Lévy processes F(ω)=E[F]+ ∑ m⩾1 ∫ 0 T E[D t (m)F| F t]♢ Y • t (m) dt. Here E[ F] is the generalized expectation, the operators D t ( m) F, m⩾1 are (generalized) Malliavin derivatives, ♢ is the Wick product and for all m⩾1 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 Lévy process. The formula holds for all F∈ G ∗⊃L 2(μ) , where G ∗ is a space of stochastic distributions and μ 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 Lévy processes.