MPS-RR 1998-11
July 1998

The Malliavin Derivative of Generalized Random Variables

by:

Fred Espen Benth

Abstract

We extend the Malliavin derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative has to be interpreted in the sense of distributions. Many of the properties of the standard Malliavin derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the extended Malliavin derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Malliavin derivative of a class of stochastic partial differential equations of Wick type.

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This paper has now been published in Has appeared in revised form as The Gross Derivative and Generalized Random Variables. Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 2 (3), 381-396 (1999)