Randomization method and backward SDEs for optimal control of partially observed path-dependent stochastic systems

TL;DR

This paper introduces a unified framework for stochastic control of partially observed, path-dependent systems without non-degeneracy assumptions, using a novel randomization method linked to backward SDEs to characterize optimal control solutions.

Contribution

It develops a general methodology employing the randomization method to handle complex control problems with partial observation and path dependence, establishing equivalence with a randomized control problem and linking it to backward SDEs.

Findings

Proves equivalence between primal and randomized control problems.

Establishes a duality with backward SDEs for control characterization.

Applicable to classical Gaussian noise control problems.

Abstract

We consider a unifying framework for stochastic control problem including the following features: partial observation, path-dependence (both with respect to the state and the control), and without any non-degeneracy condition on the stochastic differential equation (SDE) for the controlled state process, driven by a Wiener process. In this context, we develop a general methodology, refereed to as the randomization method, studied in [23] for classical Markovian control under full observation, and consisting basically in replacing the control by an exogenous process independent of the driving noise of the SDE. Our first main result is to prove the equivalence between the primal control problem and the randomized control problem where optimization is performed over change of equivalent probability measures affecting the characteristics of the exogenous process. The randomized problem turns out to be associated by duality and separation argument to a backward SDE, which leads to the so-called randomized dynamic programming principle and randomized equation in terms of the path-dependent filter, and then characterizes the value function of the primal problem. In particular, classical optimal control problems with partial observation affected by non-degenerate Gaussian noise fall within the scope of our framework, and are treated by means of an associated backward SDE.