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Martingale representation for Poisson processes with applications to minimal variance hedging

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  • Last, Günter
  • Penrose, Mathew D.

Abstract

We consider a Poisson process [eta] on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere 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 and a [sigma]-finite measure on another space . 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 Itô 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

  • Last, Günter & Penrose, Mathew D., 2011. "Martingale representation for Poisson processes with applications to minimal variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1588-1606, July.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1588-1606
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    References listed on IDEAS

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    1. Fred Espen Benth & Giulia Di Nunno & Arne Løkka & Bernt Øksendal & Frank Proske, 2003. "Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 55-72, January.
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    Cited by:

    1. Marcus C. Christiansen, 2018. "A martingale concept for non-monotone information in a jump process framework," Papers 1811.00952, arXiv.org, revised Jan 2021.
    2. Marcus C. Christiansen, 2021. "Time-dynamic evaluations under non-monotone information generated by marked point processes," Finance and Stochastics, Springer, vol. 25(3), pages 563-596, July.

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