Optimal Variational Principle for Backward Stochastic Control Systems Associated with L\'{e}vy Processes

TL;DR

This paper develops a variational framework for optimal control of backward stochastic differential equations driven by Lévy processes, providing necessary and sufficient conditions for optimality and applying it to linear-quadratic problems.

Contribution

It introduces a novel variational principle for backward stochastic control systems associated with Lévy processes, extending existing methods to more general stochastic dynamics.

Findings

Derived necessary and sufficient optimality conditions using convex variation and duality.

Characterized the optimal control for backward linear-quadratic problems via stochastic Hamilton systems.

Extended control theory to include Lévy process-driven backward stochastic systems.

Abstract

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel's martingales and an independent multi-dimensional Brownian motion, where Teugel's martingales are a family of pairwise strongly orthonormal martingales associated with L\'{e}vy processes (see Nualart and Schoutens \cite{NuSc}). We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (called backward linear-quadratic problem, or BLQ problem for short) is discussed and characterized by stochastic Hamilton system.