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)|X1]. 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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Mathematics Subject Classification:
60H07, 65C05, 49-00
JEL Classification:
G10, C10
The authors gratefully acknowledge for the comments raised by an anonymous referee, which helped understanding the existence result of Sect. [4.2] of this paper.
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Bouchard, B., Ekeland, I. & Touzi, N. On the Malliavin approach to Monte Carlo approximation of conditional expectations. Finance and Stochastics 8, 45–71 (2004). https://doi.org/10.1007/s00780-003-0109-0
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DOI: https://doi.org/10.1007/s00780-003-0109-0