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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H07 Stochastic calculus of variations and the Malliavin calculus
- 65C05 Monte Carlo methods
- 49-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

**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