An Optimal Control Derivation of Nonlinear Smoothing Equations

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton's equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen's minimum energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin

Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition

This paper is devoted to present a method of proving verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term. The value function is assumed to be continuous in time and once differentiable in the space variable ($C^{0,1}$) instead of once differentiable in time and twice in space ($C^{1,2}$), like in the classical results. The results are obtained using a time dependent Fukushima - Dirichlet decomposition proved in a companion paper by the same authors using stochastic calculus via regularization. Applications, examples and comparison with other similar results are also given.Comment: 34 pages. To appear: Stochastic Processes and Their Application

Weak Dirichlet processes with a stochastic control perspective

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case that the value function is assumed to be continuous in time and once differentiable in the space variable ($C^{0,1}$) instead of once differentiable in time and twice in space ($C^{1,2}$), like in the classical results. For this purpose, the replacement tool of the It\^{o} formula will be the Fukushima-Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale $M$ plus an adapted process $A$ which is orthogonal, in the sense of covariation, to any continuous local martingale. The mentioned decomposition states that a $C^{0,1}$ function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition. Applications to the proof of verification theorems will be addressed in a companion paper.Comment: 22 pages. To appear: Stochastic Processes and Their Application