## Contributed Talk

### On the Malliavin approach to the computation of conditional expectations

Nizar Touzi, Bruno Bouchard

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 {\it localizing functions}. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated-variance 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.