Optimal Control Problems of Forward-Backward Stochastic Volterra Integral Equations

TL;DR

This paper develops Pontryagin maximum principles for optimal control problems involving forward-backward stochastic Volterra integral equations, introducing a duality principle that advances the theoretical understanding of these complex stochastic systems.

Contribution

It introduces a novel duality principle for linear backward stochastic and Fredholm-Volterra integral equations, enabling new maximum principles for FBSVIE control problems.

Findings

Established a general duality principle for linear backward stochastic equations.

Proved Pontryagin maximum principles for FBSVIE control problems.

Enhanced theoretical tools for stochastic Volterra integral equations.

Abstract

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs in short) are formulated and studied. A general duality principle is established for linear backward stochastic integral equation and linear stochastic Fredholm-Volterra integral equation with mean-field. With the help of such a duality principle, together with some other new delicate and subtle skills, Pontryagin type maximum principles are proved for two optimal control problems of FBSVIEs.