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