Infinite horizon backward stochastic Volterra integral equations and discounted control problems

TL;DR

This paper studies infinite horizon backward stochastic Volterra integral equations (BSVIEs), establishing existence, uniqueness, and duality results, and applies these to stochastic control problems with discounted costs, including fractional and integro-differential equations.

Contribution

It extends finite horizon BSVIE results to the infinite horizon case, providing new theoretical tools and optimality conditions for related control problems.

Findings

Proved existence and uniqueness of solutions in weighted L^2 spaces.

Established a duality principle between linear SVIEs and BSVIEs.

Derived Pontryagin's maximum principle for infinite horizon control problems.

Abstract

Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted $L^2$-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted $L^2$-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin's maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.