A sufficient condition for optimal control problem of fully coupled forward-backward stochastic systems with jumps: A state-constrained control approach

Abstract

We study the stochastic optimal control problem for fully coupled forward-backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state-constrained forward stochastic control problem (referred to as (Figure presented.)) that includes additional (possibly unbounded) control variables. Then (Figure presented.) is solved via the backward reachability analysis, by which the value function of (Figure presented.) is expressed as the zero-level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as (Figure presented.)). Unlike (Figure presented.)), (Figure presented.) is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for (Figure presented.) is the unique viscosity solution to the associated integro-type Hamilton-Jacobi-Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve (Figure presented.) to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then (Figure presented.) is solved by characterizing the zero-level set of the value function of (Figure presented.), from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear-quadratic problem for fully coupled FBSDEs with jumps are also presented.