Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

On the Malliavin approach to Monte Carlo approximation of conditional expectations


Code(s) de Classification MSC:

Résumé: 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.

Mots Clés: Monte Carlo ; Malliavin calculus ; calculus of variations

Date: 2002-02-12

Prépublication numéro: PMA-709

Pdf file : PMA-709.pdf