Martingale representation for Poisson processes with applications to minimal variance hedging

TL;DR

This paper derives a new explicit martingale representation formula for Poisson processes with applications to minimal variance hedging in financial markets, extending previous results to more general settings.

Contribution

It provides a novel Clark-Ocone type formula for Poisson processes with general intensity measures, enabling explicit minimal variance hedging strategies.

Findings

Derived a new Clark-Ocone type formula for general Poisson processes.

Extended the representation to independent random measures of jump type.

Explicitly characterized minimal variance hedging in complex financial models.

Abstract

We consider a Poisson process $\eta$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $\eta$), which was previously known only in the special case, when $\lambda$ is the product of Lebesgue measure on $\R_+$ and a $\sigma$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of It\^o of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.