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Finite Difference Methods for the Weak Solutions of the Kolmogorov Equations for the Density of Both Diffusion and Conditional Diffusion Processes,
BROWN UNIV PROVIDENCE R I LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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The paper treats the problem of obtaining numerical solutions to the Fokker-Plank equation for the density of a diffusion, and for the conditional density, given certain white noise corrupted observations. These equations generally have a meaning only in the weak sense the basic assumptions on the diffusion are that the coefficients are bounded, and uniformly continuous, and that the diffusion has a unique solution in the same of multivariate distributions. It is shown that, if the finite difference approximations are carefully but naturally chosen, then the finite difference solutions to the formal adjoints, yield immediately, a sequence of approximations which converge weakly to the weak sense solution to the Fokker-Plank equation conditional or not, as the difference intervals to to zero. Modified author abstract
APPROVED FOR PUBLIC RELEASE