Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

TL;DR

This paper develops a novel approach to partial observation stochastic control problems by characterizing the value function as a viscosity solution to a new fully nonlinear PDE on Wasserstein space, without requiring non-degeneracy conditions.

Contribution

It introduces a randomized dynamic programming principle and a characterization of the value function as a unique viscosity solution to a new PDE on Wasserstein space, applicable to non-Gaussian models.

Findings

Proves a DPP for partially observed control problems using control randomization.

Characterizes the value function as a viscosity solution to a new PDE on Wasserstein space.

Provides an explicit solution for a linear quadratic model with partial observations.

Abstract

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model.