Optimal control and zero-sum game subject to differential equations with Liu processes and random matrices

This paper presents a differential equation including both random matrices and a Liu process. Then we demonstrate that the solution to this equation exists and is unique. Under the framework of chance theory, problems of optimal control and two-person zero-sum game subject to differential equations are considered. An equation of optimality is provided for solving a problem of optimal control. Then equilibrium equations are proposed to identify the saddle-point of a two-person zero-sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including both random matrices and multiple Liu processes. Finally, we utilize the acquired theoretical results to analyze a portfolio selection game problem.