On the Malliavin approach to Monte Carlo approximation of conditional expectations

Bruno Bouchard (), Ivar Ekeland () and Nizar Touzi ()

Finance and Stochastics, 2004, vol. 8, issue 1, 45-71

Abstract: Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X 2 )|X 1 ]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle. Copyright Springer-Verlag Berlin/Heidelberg 2004

Keywords: Monte Carlo; Malliavin calculus; calculus of variations (search for similar items in EconPapers)
Date: 2004
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Citations: View citations in EconPapers (17)

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DOI: 10.1007/s00780-003-0109-0

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