Stochastic maximum principle for time-changed forward-backward stochastic control problem with L\'evy noise

TL;DR

This paper develops a stochastic maximum principle for control problems involving time-changed forward-backward stochastic differential equations with Le9vy noise, modeling phenomena like trapping and subdiffusion.

Contribution

It introduces a new maximum principle for systems with time-changed processes and Le9vy noise, including necessary and sufficient optimality conditions.

Findings

Derived duality transformation and adjoint equations.

Established necessary and sufficient optimality conditions.

Applied results to a cash management problem.

Abstract

This paper establishes a stochastic maximum principle for optimal control problems governed by time-changed forward-backward stochastic differential equations with L\'evy noise. The system incorporates a random, non-decreasing operational time (the inverse of an $\alpha$-stable subordinator) to model phenomena like trapping events and subdiffusion. Using a duality transformation and the convex variational method, we derive necessary and sufficient conditions for optimality, expressed through a novel set of adjoint equations. Finally, the theoretical results are applied to solve an explicit cash management problem under stochastic recursive utility.