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

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.