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Martingale representation theorem in infinite dimensions. (English) Zbl 1058.60028

Summary: We prove a martingale representation theorem for Hilbert space valued martingales, adapted to filtration generated by a given Wiener process \(W\) on another separable Hilbert space \(H\). Two cases are considered: first when \(W\) is cylindrical, and second when \(W\) is a genuine \(Q\)-Wiener process on \(H\). A Clark-Ocone theorem is derived in this setting to give an explicit form for the integrand in this theorem.

MSC:

60G46 Martingales and classical analysis
60H07 Stochastic calculus of variations and the Malliavin calculus
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
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