Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions

TL;DR

This paper develops a novel approach for solving optimal control problems involving forward-backward stochastic Volterra integral equations with non-convex control regions, using set-valued analysis and duality principles.

Contribution

It introduces a new method to handle non-convex control regions without spike variations, deriving duality principles and first-order optimality conditions for FBSVIEs.

Findings

Derived a duality principle extending previous results.

Established first-order necessary optimality conditions.

Simplified the control problem by requiring only one adjoint system.

Abstract

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to treat the non-convexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our stochastic control systems. A duality principle between linear backward stochastic Volterra integral equations and linear stochastic Fredholm-Volterra integral equations with conditional expectation are derived, which extends and improves the corresponding results in [25], [30]. Some first order necessary optimality conditions for optimal controls of FBSVIEs are established. In contrast with existed common routines to treat the non-convexity of stochastic control problems, here only one adjoint system and one-order differentiability requirements of the coefficients are needed.