Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.

Key Words:

• Levy process
• Poisson random measure
• Martingale representation
• Hedging

AMS Subject Classification:

• Primary 60H07
• Secondary 91B28, 60G51

Acknowledgments

I would like to thank F. E. Benth for suggestions and helpful comments. Thanks also to G. Us and D. Nualart for helpful remarks. Part of this work was carried out while visiting the MaPhySto centre in Aarhus. I appreciate the hospitality provided by O. Barndorff-Nielsen and financial support received from NorFa during my stay at MaPhySto. I also acknowledge financial support, grant 134228/432, provided by the Norwegian research council (NFR).