Causal feedback strategies for controlled stochastic Volterra systems: a unified treatment

TL;DR

This paper develops a unified framework for linear quadratic control of stochastic Volterra integral equations, introducing a novel class of optimal causal feedback strategies characterized by a new Riccati system, extending existing theories.

Contribution

It introduces a new Riccati-based approach and a fundamental function space for causal feedback control in stochastic Volterra systems, unifying and extending prior results.

Findings

New Riccati system characterizes optimal feedback.

Causal feedback structure differs from classical state feedback.

Extension of control theory to various Volterra and integro-differential equations.

Abstract

This paper is concerned with a unified treatment of linear quadratic control problem for stochastic Volterra integral equations (SVIEs), motivated by the various approaches and scattered results in the existing literature. A novel class of optimal causal feedback strategy is introduced and characterized by means of a new Riccati system. To this end, a fundamental function space and an appropriate multiplicative rule among functions are defined for the first time. In contrast with the existing works, our unified treatment not only provides a new approach, but also extends or improves the known conclusions in stochastic differential equations, convolution SVIEs, stochastic Volterra integro-differential equations (VIDEs), deterministic VIEs, deterministic VIDEs. In addition, an interesting phenomenon is reveal by the current study: for SVIEs the conventional structure of state feedback is replaced by a suitable causal form, and the original state process no longer plays indispensable role in the feedbacks while an auxiliary state process does.