General maximum principles for optimal control problems of stochastic Volterra integral equations

TL;DR

This paper develops maximum principles for optimal control of stochastic Volterra integral equations, introducing new methodologies that avoid reliance on Itô calculus and second-order adjoint equations, revealing novel features of the control problem.

Contribution

It establishes Pontryagin's maximum principle for stochastic Volterra integral equations using a new approach that introduces two adjoint processes, differing from the SDE case.

Findings

Necessary conditions for optimal controls are derived via spike variation.

The methodology avoids Itô formula and second-order adjoint equations.

Two adjoint processes are required, revealing new features in the Volterra setting.

Abstract

Optimal control problems of forward stochastic Volterra integral equations (SVIEs) are formulated and studied. When control region is arbitrary subset of Euclidean space and control enters into the diffusion, necessary conditions of Pontryagin's type for optimal controls are established via spike variation. Our conclusions naturally cover the analogue of stochastic differential equations (SDEs), and our developed methodology drops the reliance on It\^o formula and second-order adjoint equations. Some new features, that are concealed in the SDEs framework, are revealed in our situation. For example, instead of using second-order adjoint equations, it is more appropriate to introduce second-order adjoint processes. Moreover, the conventional way of using one second-order adjoint equation is inadequate here. In other words, two adjoint processes, which just merge into the solution of second-order adjoint equation in SDEs situation, are actually required and proposed in our setting.