Robust continuous-time smoothers-without two-sided stochastic integrals

We consider the problem of fixed-interval smoothing of a continuous-time partially observed nonlinear stochastic dynamical system. Existing results for such smoothers require the use of two sided stochastic calculus. The main contribution of this paper is to present a robust formulation of the smoothing equations. Under this robust formulation, the smoothing equations are non-stochastic parabolic partial differential equations (with random coefficients)-and hence the technical machinery associated with two sided stochastic calculus is not required. Furthermore, the robust smoothed state estimates are locally Lipschitz in the observations-which is useful for numerical simulation. As examples, finite dimensional robust versions of the hidden Markov model smoothers are derived-these finite dimensional smoothers do not involve stochastic integrals.