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 processesF(ω)=E[F]+∑m⩾1∫0T E[Dt(m)F|Ft]♢Y•t(m) dt.
Here E[F] is the generalized expectation, the operators Dt(m)F,m⩾1 are (generalized) Malliavin derivatives, ♢ is the Wick product and for all m⩾1Y•t(m) is the white noise of power jump processes Yt(m). In particular, Y•t(1) is the white noise of the Lévy process. The formula holds for all F∈G∗⊃L2(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.
