Explicit Form and Path Regularity of Martingale Representations

Jin Ma (), Philip Protter () and Jianfeng Zhang ()
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Jin Ma: Purdue University, Department of Mathematics
Philip Protter: Purdue University, Departments of Mathematics and Statistics
Jianfeng Zhang: Purdue University, Department of Mathematics

A chapter in Lévy Processes, 2001, pp 337-360 from Springer

Abstract: 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.

Keywords: Conditional Variation; Dominate Convergence Theorem; Price Process; Contingent Claim; Incomplete Market (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0197-7_15

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DOI: 10.1007/978-1-4612-0197-7_15

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