Near-optimal control problems for forward-backward regime-switching systems^*

School of Mathematics, Shandong University, Jinan 250100, P.R. China.

^** Corresponding author: wuzhen@sdu.edu.cn

Received: 11 September 2018
Accepted: 4 April 2020

Abstract

This paper investigates the near-optimality for a class of forward-backward stochastic differential equations (FBSDEs) with continuous-time finite state Markov chains. The control domains are not necessarily convex and the control variables do not enter forward diffusion term. Some new estimates for state and adjoint processes arise naturally when we consider the near-optimal control problem in the framework of regime-switching. Inspired by Ekeland’s variational principle and a spike variational technique, the necessary conditions are derived, which imply the near-minimum condition of the Hamiltonian function in an integral sense. Meanwhile, some certain convexity conditions and the near-minimum condition are sufficient for the near-optimal controls with order ε^1/2. A recursive utility investment consumption problem is discussed to illustrate the effectiveness of our theoretical results.