Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

TL;DR

This paper develops a causal state feedback representation for linear-quadratic optimal controls of stochastic Volterra equations using path-dependent Riccati equations, enabling decoupling of the optimality system.

Contribution

It introduces a path-dependent Riccati equation and a decoupling method for the optimality system of stochastic Volterra equations, providing a novel causal feedback control framework.

Findings

Existence and uniqueness of solutions to the path-dependent Riccati equation.

Decoupling of the optimality system via Riccati equations.

Reduction to Markovian feedback when control affects only the diffusion term.

Abstract

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE, for short) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a Type-II backward SVIE (BSVIE, for short). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a Type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding Type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback.