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Martingale representation for Poisson processes with applications to minimal variance hedging

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  • Guenter Last
  • Mathew D. Penrose

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.

Suggested Citation

  • Guenter Last & Mathew D. Penrose, 2010. "Martingale representation for Poisson processes with applications to minimal variance hedging," Papers 1001.3972, arXiv.org.
  • Handle: RePEc:arx:papers:1001.3972
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    File URL: http://arxiv.org/pdf/1001.3972
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