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

Mingshang Hu, Shaolin Ji+1 more|arXiv (Cornell University)|Dec 22, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance19 references3 citations

TL;DR

This paper extends the global stochastic maximum principle to fully coupled forward-backward stochastic systems with nonconvex control domains by introducing two general assumptions that ensure the applicability of Hu et al.'s approach. It establishes the maximum principle under weaker conditions than the original weakly coupled assumption, proving existence and estimates for variational equations and adjoint processes, and generalizes the state space to multi-dimensional case.

ABSTRACT

Hu et. al 2018 studied a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. By assuming a weakly coupled condition, they established an approach to obtain the first-order, second-order variational equations and the adjoint equations for the states X, Y and Z and deduced the global maximum principle. But it is well known that there are several different conditions such as monotonicity condition, weakly coupled condition and other conditions which can guarantee the existence and uniqueness of the solution to fully coupled FBSDEs. In this note, to overcome the limitations of assuming a specific condition, we propose two kinds of assumptions which can guarantee that the approach developed in Hu et. al 2018 is still applicable. Under these two kinds of assumptions, we obtain the global stochastic maximum principle.

Motivation & Objective

• To address the limitation of requiring a specific condition like weakly coupled or monotonicity for the global stochastic maximum principle in fully coupled forward-backward stochastic control systems.
• To identify minimal, general assumptions that preserve the validity of Hu et al.'s approach for deriving first- and second-order variational equations and adjoint equations.
• To prove the global stochastic maximum principle holds under these generalized conditions, including multi-dimensional state processes.
• To establish $L^p$-estimates for solutions of FBSDEs and their variational equations under the new assumptions.
• To generalize the framework beyond the original scalar state assumption to multi-dimensional state variables.

Proposed method

• Propose two new sets of assumptions that replace the restrictive weakly coupled condition: one based on bounded first-order adjoint solutions, the other on linear dependence of diffusion coefficient $\sigma$ on $z$.
• Use spike variation method to derive necessary conditions for optimality, leveraging the variational equations for $X$, $Y$, and $Z$.
• Establish $L^p$-estimates for the solution of the FBSDE and its variational equations under the new assumptions, ensuring stability and integrability.
• Apply the Ekeland variational principle indirectly by ensuring the existence and uniqueness of solutions to the FBSDE and adjoint equations.
• Generalize the state process $X$ from scalar to multi-dimensional case, maintaining the structure of the fully coupled FBSDE.
• Use Itô's formula and Gronwall-type inequalities to derive bounds on the solutions of variational equations, relying on the new assumptions to control growth and integrability.

Experimental results

Research questions

• RQ1 Can the global stochastic maximum principle for fully coupled FBSDEs be established without assuming a specific structural condition like weakly coupled or monotonicity?
• RQ2 What minimal assumptions on the coefficients and solution properties are sufficient to ensure the validity of the variational approach in Hu et al. [4]?
• RQ3 How can $L^p$-estimates for the variational equations be derived under general assumptions rather than specific coefficient structures?
• RQ4 Under what conditions does the first-order adjoint equation admit a unique bounded solution, and how does this affect the maximum principle?
• RQ5 Can the framework be extended to multi-dimensional state processes while preserving the optimality conditions?

Key findings

• The global stochastic maximum principle is established under two new general assumptions, removing the need for restrictive structural conditions like weakly coupled or monotonicity.
• The first assumption—unique solution to FBSDE and bounded solution to first-order adjoint equation—ensures $L^p$-estimates for the state and variational processes.
• The second assumption—linear $\sigma$ in $z$, unique FBSDE solution, $L^p$-estimates, and unique first-order adjoint solution—also ensures the required estimates and validity of the maximum principle.
• The paper proves that under both assumption sets, the first- and second-order variational equations satisfy the necessary estimates for deriving the maximum principle.
• The state process $X$ is generalized from scalar to multi-dimensional case, broadening the applicability of the framework.
• The results are robust under nonconvex control domains, resolving an open problem in Peng [16] regarding first-order variational equations in nonconvex settings.