Linear Quadratic Pareto Game Problem: A Case of Deterministic Controller

In this paper, we study a kind of linear quadratic Pareto game problem in stochastic system. The biggest difference from the existing literature, is that the control is described by deterministic time function. This kind of game is inspired by some national policy arising from government and can be used to present the players' action. By applying the weighted sum method, we could transform the Pareto game into an optimal control problem and by virtue of Pontryagin maximum principle, we derive all Pareto efficient strategies. Moreover, we conclude that the optimal state is characterized by a linear stochastic differential equation of mean-field type, which is one of novel features in our paper. Finally, we work out one numerical example to enhance the implementability of the theoretical results.