White noise analysis for Lévy processes

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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⩾10TE[Dt(m)F|Ft]♢Yt(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⩾1Yt(m) is the white noise of power jump processes Yt(m). In particular, Yt(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.

MSC

primary 60H40
secondary 60G51
60G57
60H07
91B28

Keywords

Lévy processes
White noise
Malliavin derivatives
Chaos expansions
Generalized Clark–Haussmann–Ocone formula
Portfolios in financial markets

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