Clark-Okone type martingale representations for Poisson martingales

Martingale representations are fundamental tools of stochastic analysis. In the case of Brownian motion the Clark-Okone representation yields explicit expressions for the integrand in terms of Malliavin derivatives. A similar result is known for Poisson and Lévy processes. In this talk we will explain a general version of this representation for Poisson martingales, taken from [1]. Our first application are short proofs of the Poincare- and the FKG-inequality for Poisson processes. A second application is Wu's [2] elegant proof of a general log-Sobolev inequality for Poisson functionals. If time permits we will also discuss minimal variance hedging or some potential applications in stochastic geometry.

[1] Last, G. and Penrose, M.D. (2011). Martingale representation for Poisson processes with applications to minimal variance hedging. Stochastic Processes and their Applications 121, 1588-1606.

[2] Wu, L. (2000). A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probability Theory and Related Fields 118, 427-438.

Tomasz Schreiber's Memorial Session