On the Malliavin approach to Monte Carlo approximation of conditional expectations
Bouchard, Bruno; Ekeland, Ivar; Touzi, Nizar (2004), On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8, 1, p. 45-71. http://dx.doi.org/10.1007/s00780-003-0109-0
TypeArticle accepté pour publication ou publié
Journal nameFinance and Stochastics
MetadataShow full item record
Abstract (EN)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)|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.
Subjects / KeywordsMéthode de Monte-Carlo; Numerical Probability; calculus of variations
Showing items related by title and author.
Fournié, Eric; Lasry, Jean-Michel; Lebuchoux, Jérôme; Lions, Pierre-Louis; Touzi, Nizar (1999) Article accepté pour publication ou publié
Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier (2019) Article accepté pour publication ou publié