Pathwise solutions and multiplicative functionals in nonlinear filtering

This paper concerns the "standard" nonlinear filtering problem of calculating ??t (f) = E[f (xt) | ys\´, 0 ?? s ?? t] where xt is a Markov process, yt = h (xt) + white noise, and f is a real-valued function. The objective is to obtain a recursively calculable "robust" version of ??t (f), i.e. a version parametrized in a simple way by the observation sample path. We outline two approaches to this problem, one based on a "pathwise solution" of the Zakai (un-normalized) filtering equation and the other starting from the idea that the Kallianpur-Striebel formula represents a y-dependent multiplicative functional transformation of the signal process semigroup Tt. These two approaches are shown to be equivalent.