Online Smoothing for Diffusion Processes Observed with Noise

We introduce a methodology for online estimation of smoothing expectations for a class of additive functionals, in the context of a rich family of diffusion processes (that may include jumps) -- observed at discrete-time instances. We overcome the unavailability of the transition density of the underlying SDE by working on the augmented pathspace. The new method can be applied, for instance, to carry out online parameter inference for the designated class of models. Algorithms defined on the infinite-dimensional pathspace have been developed in the last years mainly in the context of MCMC techniques. There, the main benefit is the achievement of mesh-free mixing times for the practical time-discretised algorithm used on a PC. Our own methodology sets up the framework for infinite-dimensional online filtering -- an important positive practical consequence is the construct of estimates with the variance that does not increase with decreasing mesh-size. Besides regularity conditions, our method is, in principle, applicable under the weak assumption -- relatively to restrictive conditions often required in the MCMC or filtering literature of methods defined on pathspace -- that the SDE covariance matrix is invertible.