Optimal control of SDEs with expected path constraints and related constrained FBSDEs

TL;DR

This paper develops a stochastic maximum principle for controlling SDEs with expected path constraints, introducing a deterministic compensated process in the adjoint equation, and provides results for linear-quadratic cases.

Contribution

It introduces a new stochastic maximum principle with a deterministic compensated process and analyzes constrained reflected FBSDEs for LQ control problems.

Findings

Derived stochastic maximum principle with deterministic adjoint process

Established verification theorem for linear-quadratic control

Proved existence and uniqueness of constrained reflected FBSDEs

Abstract

In this paper, we consider optimal control of stochastic differential equations subject to an expected path constraint. The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs. In particular, the compensated process in our adjoint equation is deterministic, which seems to be new in the literature. For the typical case of linear stochastic systems and quadratic cost functionals (i.e., the so-called LQ optimal stochastic control), a verification theorem is established, and the existence and uniqueness of the constrained reflected FBSDEs are also given.