On the Malliavin approach to Monte Carlo approximation of conditional expectations

Bruno Bouchard, Ivar Ekeland, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


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 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.

Original languageEnglish (US)
Pages (from-to)45-71
Number of pages27
JournalFinance and Stochastics
Issue number1
StatePublished - Feb 2004


  • Calculus of variations
  • Malliavin calculus
  • Monte Carlo

ASJC Scopus subject areas

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


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