On the joint nonlinear filtering-smoothing of diffusion processes (Q1113865)

Let \(\{x_ t\}\) be the solution of a stochastic differential equation, and \(\{y_ t\}\) be the observation process, which is the sum of a function of \(x_ t\) and of an independent white noise. This paper gives a formula for the joint density of the law of \((x_ s,x_ t)\) (where \(0<s<t)\), given \(\{y_ r\); \(0\leq r\leq t\}\). The result is then applied to a well-known nonlinear filtering problem which possesses a finite dimensional filter.