On the optimal filtering of diffusion processes

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Summary

Let x(t) be a diffusion process satisfying a stochastic differential equation and let the observed process y(t) be related to x(t) by dy(t) = g(x(t)) + dw(t) where w(t) is a Brownian motion. The problem considered is that of finding the conditional probability of x(t) conditioned on the observed path y(s), 0≦st. Results on the Radon-Nikodym derivative of measures induced by diffusions processes are applied to derive equations which determine the required conditional probabilities.