Robust optimal control of bi-objective non-homogeneous stochastic linear quadratic system with random coefficients

This paper investigates a robust stochastic linear–quadratic optimal control (RSLQ) problem with non-homogeneous terms in state, where all coefficients of state and cost function are allowed to be random and the coefficients of cost function are also uncertain. The existence of a unique robust optimal control (OC) ${\bar{v}}^{{\lambda }^{\ast }}$ which depends on parameter ${\lambda }^{\ast }\in [0,1]$ is obtained, and $({\lambda }^{\ast },{\bar{v}}^{{\lambda }^{\ast }})$ is verified to be a saddle point of a game problem. Then the solving of Problem RSLQ is proven to be equivalent to the seeking of a global maximum point ${\lambda }^{\ast }$ of a value function $V^{\lambda }\left(x\right)$ w.r.t. $\lambda \in [0,1]$. Furthermore, for the case with one-dimensional state, we obtain the continuity of $V^{\lambda }\left(x\right)$ w.r.t. $\lambda$, which can also be seen as a stability property of value function of stochastic linear–quadratic (SLQ) problems w.r.t. parameter $\lambda$. The main challenge is the continuity dependency w.r.t. $\lambda$ of solution of an auxiliary linear backward stochastic differential equation (BSDE), which has possibly unbounded coefficients and appears in the explicit form of $V^{\lambda }\left(x\right)$. Via the estimates of bounded mean oscillation martingales (BMO martingales) and stochastic Riccati equations (SREs), along with transformation of measures, we overcome this obstacle and derive that $V^{\lambda }\left(x\right)$ is Lipschitz continuous w.r.t. $\lambda$. Estimates of two generalized SREs appeared in an SLQ problem with conic constraint are also obtained. Some numerical examples are given to further verify the validity of our theoretical analysis.