Pathwise smoothing of Markov processes with noisy observations

Abstract

Since the advent of the Feynman-Kac formula (8, 9, 171, the connections between function space integrals arising in probability theory and partial differential equations has been a field of active research. Recently, such connections have found applications in stochastic filtering theory [2, 12-14 ]. A prototypical estimation problem for random processes can be described as follows: Given a signal process (X~}IEtO,Tl and a related observation process VA,ElO,T,~ one wants to estimate E(X,/Y,, s E [0, r]) for some t E [0, r]. Depending on whether r T, the problem is referred to as a smoothing, filtering, or prediction problem. For (Xt)tE,o.r, a diffusion process, and a signal plus white noise model for ( Yl)rc,o.r,r Davis (21 has given a pathwise formulation of the filtering problem. This formulation involves expressing the filtered estimate as a function space integral that does not involve stochastic integration with respect to ( YIJIEtO.r,. Using the theory of multiplicative functionals of Markov processes, the problem is then converted to an equivalent problem of solving a deterministic p.d.e. in which the observation process acts like a parameter. This formulation has an advantage over conventional formulations in terms of being robust with respect to errors in the modelling of observation noise, in a precise sense. For a detailed discussion of this aspect, the reader is referred to the original paper of Davis [2]. In this paper, we have given a similar formulation of the smoothing problem. The claim that this formulation is robust with respect to the errors in modelling the observation noise can be justified the same way as in [ 21. hence the arguments are not repeated. The techniques used here are quite different from those used in [2] and are more in the spirit of [S, 6] (see also ] 10, pp. 50-521). For the special case of t = T. this approach provides an alternative derivation for some of the results in [ 21. The smoothing problem is much more difftcult than the filtering problem