Filtering of absorbing and reflecting Brownian motions (Q1821079)

The solutions of two nonlinear filtering problems are made explicit following a Beneš' technique. In these two problems, the observation of the signal is corrupted by an independent white noise and the signal process is a one-dimensional Brownian motion which is either stopped (first problem) or reflected (second problem) when it hits 0. Nonlinear filtering is reduced in these two cases to second-order boundary value problems and formulas involving Volterra equations are given for solving them; in the second problem, two different methods are described.