[Paper Review] A global stochastic maximum principle for fully coupled forward-backward stochastic systems

Mingshang Hu, Shaolin Ji+1 more|arXiv (Cornell University)|Mar 6, 2018

Climate Change Policy and Economics4 citations

TL;DR

This paper establishes a global stochastic maximum principle for fully coupled forward-backward stochastic control systems with nonconvex control domains by introducing a novel decoupling approach using a quadratic backward stochastic differential equation (BSDE) as the adjoint equation. The method reveals hidden relations among first-order Taylor expansion terms, enabling derivation of a complete maximum principle that includes a new term, overcoming limitations in prior methods when diffusion coefficients depend on controls.

ABSTRACT

We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear FBSDEs. Inspired by Hu (Hu, Probability, Uncertainty and Quantitative Risk, 2(1) (2017):pp 1-20), we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE. By revealing the relations among the terms of the first-order Taylor's expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated.

Motivation & Objective

• To address the open problem of deriving a global maximum principle for stochastic recursive optimal control with nonconvex control domains.
• To overcome the failure of classical first-order expansion methods in stochastic systems where diffusion terms depend on controls.
• To develop a new decoupling technique for fully coupled linear FBSDEs arising from variational equations.
• To eliminate unknown parameters in optimality conditions, as present in Ekeland variational principle-based approaches.
• To establish a complete global maximum principle that includes a novel term not found in prior local or classical formulations.

Proposed method

• Introduce an adjoint equation that is a quadratic BSDE, derived from the first-order variational equations of the FBSDE system.
• Reveal and exploit hidden relations among terms in the first-order Taylor expansion of the state and adjoint processes.
• Decouple the fully coupled linear FBSDEs by expressing the variation of Z(t) in terms of p(t), δσ(t), and X₁(t), using a new decomposition.
• Apply Itô’s formula to the decoupled system to derive the optimality condition and verify the solution structure.
• Use the Ekeland variational principle as a foundation but eliminate unknown parameters through the new adjoint structure.
• Prove uniqueness of the adjoint process (p(·), q(·)) under smallness conditions on γ₂(·) and integrability assumptions on coefficients.

Experimental results

Research questions

• RQ1 Can a global stochastic maximum principle be derived for fully coupled forward-backward stochastic control systems when the control domain is nonconvex and diffusion coefficients depend on controls?
• RQ2 What is the role of the first-order Taylor expansion terms in the variational equations, and how can their interdependence be exploited to achieve decoupling?
• RQ3 How can the unknown parameters in the Ekeland variational principle approach be eliminated to yield a closed-form maximum principle?
• RQ4 What is the structure of the adjoint equation in this fully coupled setting, and how does it differ from classical linear adjoint BSDEs?
• RQ5 Under what conditions is the solution to the adjoint FBSDE unique, ensuring the validity of the maximum principle?

Key findings

• A new global stochastic maximum principle is derived that includes a previously unaccounted-for term, extending beyond classical and local maximum principles.
• The first-order variational equations are fully coupled linear FBSDEs, but their structure allows decoupling via relations between Y₁(t), Z₁(t), and the adjoint process (p(t), q(t)).
• The adjoint equation is a quadratic BSDE, which enables the derivation of the maximum principle without relying on spike variation or Ekeland’s principle with unknown parameters.
• Uniqueness of the adjoint process (p(·), q(·)) is proven under smallness conditions on γ₂(·) and integrability assumptions, ensuring the principle’s consistency.
• The method successfully overcomes the technical obstacle of insufficient regularity/integrability of Z(·) in second-order expansions, resolving a key difficulty in prior work.
• Applications to stochastic linear-quadratic control problems are demonstrated, showing the principle’s practical relevance and validity.