A maximum principle for forward-backward stochastic Volterra integral equations and applications in finance

TL;DR

This paper develops a stochastic maximum principle for forward-backward stochastic Volterra integral equations, introduces a linear quadratic control problem, and applies these results to risk minimization and portfolio optimization in finance.

Contribution

It formulates a new maximum principle for FBSVIEs, proposes a simplified method for solving BSVIEs, and applies these to financial risk management and portfolio optimization.

Findings

Established a stochastic maximum principle for FBSVIEs.

Proposed a new method for the unique solvability of BSVIEs.

Derived closed-form solutions for optimal portfolios in specific cases.

Abstract

This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.