Clark--Ocone formula and variational representation for Poisson functionals

TL;DR

This paper establishes a Clark--Ocone formula for bounded Poisson functionals and derives a variational representation for their Laplace transforms, advancing large deviations theory for Poisson processes.

Contribution

It introduces a Clark--Ocone formula for bounded Poisson functionals and proves a variational representation for their Laplace transforms, confirming a conjecture by Dupuis and Ellis.

Findings

Clark--Ocone formula for bounded Poisson functionals proved

Variational representation formula for Laplace transforms established

Supports large deviations analysis for Poisson processes

Abstract

In this paper we first prove a Clark--Ocone formula for any bounded measurable functional on Poisson space. Then using this formula, under some conditions on the intensity measure of Poisson random measure, we prove a variational representation formula for the Laplace transform of bounded Poisson functionals, which has been conjectured by Dupuis and Ellis [A Weak Convergence Approach to the Theory of Large Deviations (1997) Wiley], p. 122.