Linear-quadratic stochastic Volterra controls I: Causal feedback strategies

TL;DR

This paper develops a new framework for causal feedback strategies in stochastic Volterra control problems, establishing existence, uniqueness, and duality principles for solutions involving novel integral equations.

Contribution

It introduces the concept of causal feedback strategies for controlled SVIEs and proves their well-posedness and duality using new integral equations.

Findings

Existence and uniqueness of causal feedback solutions for controlled SVIEs

Development of Type-II extended backward stochastic Volterra integral equations

A duality principle and representation formula for quadratic functionals

Abstract

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear-quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov--Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.