Title:

The problem of nonlinear filtering

Stochastic filtering theory studies the problem of estimating an unobservable 'signal' process X given the information obtained by observing an associated process Y (a 'noisy' observation) within a certain time window [0, t]. It is possible to explicitly describe the distribution of X given Y in the setting of linear/guassain systems. Outside the realm of the linear theory, it is known that only a few very exceptional examples have explicitly described posterior distributions. We present in detail a class of nonlinear filters (Bene filters) which allow explicit formulae. Using the explicit expression of the Laplace transform of a functional of Brownian motion we give a direct computation of the unnormalized conditional density of the signal of the Bene filter and obtain the formula for the normalized conditional density of X for two particular filters. In the case in which X is a diffusion process and Y is given by the equation dY_{t} = dh(s,X_{s})ds + dW_{t}, where W is a Brownian motion independent of X, Y_{0} = 0 and h satisfies certain conditions, the evolution of the conditional distribution of X is described by 2 stochastic partial differential equations: a linear equation  the Zakai equation  which describes the evolution of an unnormalised version of the condition distribution of X and a nonlinear equation  the Kushner  Stratonovitch equation  which describes the evolution of the conditional distribution of X itself. We construct several measure valued processes, associated with the two equations, whose values give the conditional distribution of X (in the first case unnormalised). We do this by means of converging sequences of branching particle systems. The particles evolve independently, moving with the same law as X, and branch according to a mechanism that depends on the their locations and the observation Y. The result is a cloud of paths, with those surviving to the current time providing an estimate for the conditional distribution of X.
