An Introduction to Nonlinear Filtering

In this paper we provide an introduction to nonlinear filtering from two points of view: the innovations approach and the approach based upon an unnormalized conditional density. The filtering problem concerns the estimation of an unobserved stochastic process xt given observations of a related process yt; the classic problem is to calculate, for each t, the conditional distribution of xt given ys, 0 ≤ s ≤ t. First, a brief review of key results on martingales and markov and diffusion processes is presented. Using the innovations approach, stochastic differential equations for the evolution of conditional statistics and of the conditional measure of xt given ys, 0 ≤ s ≤ t are given; these equations are the analogs for the filtering problem of the kolmogorov forward equations. Several examples are discussed. Finally, a less complicated evolution equation is derived by considering an “unnormalized” conditional measure.

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