Mean-field stochastic linear–quadratic optimal control with Markov jump parameters

This paper considers a class of mean-field stochastic linear–quadratic optimal control problems with Markov jump parameters. The new feature of these problems is that means of state and control are incorporated into the systems and the cost functional. Based on the modes of Markov chain, the corresponding decomposition technique of augmented state and control is introduced. It is shown that, under some appropriate conditions, there exists a unique optimal control, which can be explicitly given via solutions of two generalized difference Riccati equations. A numerical example sheds light on the theoretical results established.