On a multiplicative functional transformation arising in nonlinear filtering theory

SummaryThis paper concerns the nonlinear filtering problem of calculating “estimates” E[f(xt)¦y s, s≦t] where {xt} is a Markov process with infinitesimal generator A and {yt} is an observation process given by dyt=h(xt)dt +dwtwhere {wt} is a Brownian motion. If h(xt) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup Ts,tyobtained by a certain y-dependent multiplicative functional transformation of the signal process {xt}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is Atyf=ey(t)h(A− 1/2h2)(e−y(t)hf).