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Abstract
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We introduce a probabilistic numerical scheme for fully nonlinear parabolic PDEs, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. We mention 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. We also mention the result which proves the convergence of the latter approximation scheme, and derives 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. We also provide a generalization to Elliptic PDEs and nonlocal fully nonlinear PDEs (Integro-differential PDEs). |
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