Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations

TL;DR

This paper investigates stochastic optimal control problems involving fully coupled FBSDEs, establishing the deterministic nature of value functions, their viscosity solutions to HJB equations, and addressing complexities when diffusion coefficients depend on solution components.

Contribution

It introduces a new method to prove the value functions satisfy the DPP and are viscosity solutions, handling cases where diffusion coefficients depend on the solution's Z component.

Findings

Value functions are deterministic and satisfy DPP.

Existence and uniqueness of solutions for fully coupled FBSDEs under certain conditions.

The associated HJB equations are linked with algebraic equations, with solutions characterized.

Abstract

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We study two cases of diffusion coefficients $\sigma$ of FSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle (DPP), and are viscosity solutions to the associated generalized Hamilton-Jacobi-Bellman (HJB) equations. The associated generalized HJB equations are related with algebraic equations when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE and doesn't depend on the control. For this we adopt Peng's BSDE method, and so in particular, the notion of stochastic backward semigroup in [16]. We emphasize that the fact that $\sigma$ also depends on $Z$ makes the stochastic control much more complicate and has as consequence that the associated HJB equation is combined with an algebraic equation, which is inspired by Wu and Yu [19]. We use the continuation method combined with the fixed point theorem to prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove some new basic estimates for fully coupled FBSDEs under the monotonic assumptions. In particular, we prove under the Lipschitz and linear growth conditions that fully coupled FBSDEs have a unique solution on the small time interval, if the Lipschitz constant of $\sigma$\ with respect to $z$ is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.