| Affiliation: | (1) PMA and CREST, Université Paris VI, Boite Courrier 188, 75252 Paris, Cedex 05, France;(2) Department of Mathematical Economics, University of British Columbia, 1Z2 Vancouver V6T, Canada;(3) CREST and CEREMADE, 15 bd Gabriel Péri, 92245 Malakoff, Cedex, France |
| Abstract: | 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.Mathematics Subject Classification: 60H07, 65C05, 49-00JEL Classification: G10, C10The authors gratefully acknowledge for the comments raised by an anonymous referee, which helped understanding the existence result of Sect. [4.2] of this paper. |
