On the Malliavin approach to Monte Carlo approximation of conditional expectations
摘要：
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[sub 2])|X[sub 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.
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DOI：
10.1007/s00780-003-0109-0
被引量：
年份：
2004
Springer
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