A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

MATHEMATICAL FINANCE COLLOQUIUMSpeaker: Nizar TouziEcole Polytechnique, Paris, FranceAbstract: We suggest a probabilistic numerical scheme for fully nonlinear PDEs as a natural combination of Monte Carlo and finite differences. In the semilinear case, this scheme corresponds to the Monte Carlo scheme suggested by the representation of the solution in terms of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an
upper bound on the approximation error. Numerical experiments are performed for the approximation of the
solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.MONDAY, 3/23/09, 3:00-4:00 PM, DRB 337

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