Chaotic expansion of powers and martingale representation


Jamshidian, Farshid (2005) Chaotic expansion of powers and martingale representation.

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Abstract:This paper extends a recent martingale representation result of [N-S] for a L´evy process to filtrations generated by a rather large class of semimartingales. As in [N-S], we assume the underlying processes have moments of all orders, but here we allow angle brackets to be stochastic. Following their approach, including a chaotic expansion, and incorporating an idea of strong orthogonalization from [D], we show that the stable subspace generated by Teugels martingales is dense in the space of square-integrable martingales, yielding the representation. While discontinuities are of primary interest here, the special case of a (possibly infinite-dimensional) Brownian filtration is an easy consequence.
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