TY - JOUR

T1 - Martingale representation for Poisson processes with applications to minimal variance hedging

AU - Last, Gunter

AU - Penrose, Mathew D

PY - 2011

Y1 - 2011

N2 - We consider a Poisson process n on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure of n. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with n), which was previously known only in the special case, when is the product of Lebesgue measure on R+ and a -finite measure on another space X. 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 Ito 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.

AB - We consider a Poisson process n on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure of n. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with n), which was previously known only in the special case, when is the product of Lebesgue measure on R+ and a -finite measure on another space X. 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 Ito 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.

UR - http://www.scopus.com/inward/record.url?scp=79956208530&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1001.3972

UR - http://dx.doi.org/10.1016/j.spa.2011.03.014

U2 - 10.1016/j.spa.2011.03.014

DO - 10.1016/j.spa.2011.03.014

M3 - Article

VL - 121

SP - 1588

EP - 1606

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 7

ER -