On the Malliavin approach to Monte Carlo approximation of conditional expectations

Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X2)|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 integratedvariance minimizer among the class of separable localizing functions. For general localizing functions, we provide a PDE characterization of the optimal solution, if it exists. This allows to draw the following observation : the separable exponential function does not minimize the integrated variance, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle.