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On the Malliavin approach to Monte Carlo approximation of conditional expectations. (English) Zbl 1051.60061
Let \(X\) be a Markov process. Given \(n\) simulated paths of \(X\), the purpose of this paper is to provide a Monte Carlo estimation of the conditional expectation \(E[g(X_2)\mid X_1=x]\), i.e. the regression function of \(g(X_2)\) on \(X_1\). The Malliavin integration-by-parts formula provides a family of representations of this conditional expectation. The main contribution of the paper is the discussion of the variance reduction issue related to the family of localizing functions. The existence and uniqueness of the solution to the problem of minimization on the integrated mean-square error is proved in some class of separable functions. The solution is of an exponential form with positive parameters, characterized as the unique solution of a system of nonlinear equations. The authors also study the problem of minimizing the integrated mean-square error within a class of all localizing functions. The existence and uniqueness is proved in a suitable Sobolev space. A PDE characterization of the solution with appropriate boundary conditions is provided. An interesting observation is that separable localizing functions do not solve this equation, except for the one-dimensional case.

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
65C05 Monte Carlo methods
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