Finite difference methods for the weak solutions of the Kolmogorov equations for the density of both diffusion and conditional diffusion processes

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 sense 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 that converge weakly to the weak sense solution to the Fokker-Plank equation (conditional or not), as the difference intervals go to zero.The approximations seem very natural for this type of problem. They are related to the transition functions of a sequence of Markov chains, the measures of whose continuous time interpolations converge weakly to the (measure of) diffusion, as the difference intervals go to zero, and, hence, seem to have more physical significance than the usual (formal or not) approximations. The method is purely probabilistic and relies heavily on results of the weak convergence of measures on abstract spaces.