Optimization Under Rational Expectations: A Framework of Fully Coupled Forward-Backward Stochastic Linear Quadratic Systems

Abstract

In this paper, we propose a general modeling framework for optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs that stem from rational expectations models. We propose a new decoupling technique to obtain the optimal feedback control, which is accompanied by one kind of non-Riccati-type ordinary differential equation (ODE). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs. The obtained results make it possible to compute the control and value function. For this FBLQ problem, the optimal control should depend on the entire trajectory of the state process. Several examples are given to illustrate our results. Funding: M. Hu’s research was supported by the National Science Foundation (NSF) [Grant 11671231] and the Young Scholars Program of Shandong University [Grant 2016WLJH10]. S. Ji’s research was supported by the NSF [Grant 11571203]. X. Xue’s research was supported by “The Fundamental Research Funds of Shandong University,” the NSF [Grants 12001316 and 61907022], and the Natural Science Foundation of Shandong Province [Grant ZR2019BF015].