A Stochastic Maximum Principle for Forward-backward Stochastic Control Systems with Quadratic Generators and Sample-wise Constraints

TL;DR

This paper develops a stochastic maximum principle for forward-backward control systems with quadratic growth and sample-wise constraints, using advanced BSDE theory and variational methods, with applications in finance.

Contribution

It introduces a new SMP for systems with quadratic BSDEs and sample constraints, expanding the theoretical framework for stochastic control.

Findings

Derived a dynamic stochastic maximum principle for quadratic BSDE systems.

Applied the theory to a robust utility maximization problem with bankruptcy constraints.

Extended the control theory to include sample-wise state constraints in stochastic systems.

Abstract

This paper examines the stochastic maximum principle (SMP) for a forward-backward stochastic control system where the backward state equation is characterized by the backward stochastic differential equation (BSDE) with quadratic growth and the forward state at the terminal time is constrained in a convex set with probability one. With the help of the theory of BSDEs with quadratic growth and the bounded mean oscillation (BMO) martingales, we employ the terminal perturbation approach and Ekeland's variational principle to obtain a dynamic stochastic maximum principle. The main result has a wide range of applications in mathematical finance and we investigate a robust recursive utility maximization problem with bankruptcy prohibition as an example.