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Explicit Form and Path Regularity of Martingale Representations

In: Lévy Processes

Author

Listed:
  • Jin Ma

    (Purdue University, Department of Mathematics)

  • Philip Protter

    (Purdue University, Departments of Mathematics and Statistics
    Cornell University, ORIE)

  • Jianfeng Zhang

    (Purdue University, Department of Mathematics)

Abstract

Let X be the solution of a stochastic differential equation driven by a Wiener process and a compensated Poisson random measure, such that X is an L 2 martingale. If H = Φ(X s ; 0 ≤ s ≤ T) is in L 2, then H = α + ∫ 0 T ξ s dX s + N T , where N is an L 2 martingale orthogonal to X (the Kunita-Watanabe decomposition). We give sufficient conditions on the functional Φ such that ξ has regular paths (that is, left-continuous with right limits). In finance this has an interpretation that the risk minimizing hedging strategy of a contingent claim in an incomplete market has “smooth” regular sample paths. This means the hedging process can be approximated and the resulting approximations will converge, along the sample paths, to the risk minimal (and hence optimal) portfolio.

Suggested Citation

  • Jin Ma & Philip Protter & Jianfeng Zhang, 2001. "Explicit Form and Path Regularity of Martingale Representations," Springer Books, in: Ole E. Barndorff-Nielsen & Sidney I. Resnick & Thomas Mikosch (ed.), Lévy Processes, pages 337-360, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0197-7_15
    DOI: 10.1007/978-1-4612-0197-7_15
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