MPS-RR 1998-11

July 1998

# The Malliavin Derivative of Generalized Random Variables

by:

### Fred Espen Benth

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)*