Exact Maximum Likelihood Nonlinear Filtering Equations and Stability Properties of the Optimal Nonlinear Filter

This paper revisits the work of Mortensen (J. Optim. Theory Appl. 2, 386–394, 1968) and proposes a novel statistical method for maximum likelihood (ML) state estimation in general continuous-time stochastic systems. The merit of this method lies on the equivalence between complete-information loglikelihood of stochastic systems and the value function of associated optimal control problem of ML estimation. Distributional identity for the state estimation error is given in terms of the score function of state and observed information matrix. The dynamics of these statistical quantities are derived using Pontryagin’s maximum principle. Applying perturbation method, the distributional identity and dynamics are used to derive the exact finite-dimensional filtering equations for the ML state estimator in which the gain matrix takes a general form of Riccati equation. Stability properties of the optimal estimator are derived under posterior mean squared error and uniform observability using Lyapunov’s indirect method. Higher order implicit Runge-Kutta method is proposed for numerical solution of stochastic systems and filtering equations. Numerical examples are discussed to verify accuracy of theoretical findings.