Linear‐Quadratic Mixed Leadership Stochastic Differential Games With Overlapping Information

ABSTRACT This article studies a linear‐quadratic mixed leadership stochastic differential game with overlapping information, where each player simultaneously acts as the follower in one strategy and the leader in the other. A distinctive feature of this paper is that the information available to the two players overlaps only partially, with neither being a subset of the other. At the follower layer, the players engage in a linear‐quadratic non‐zero‐sum stochastic differential Nash game with overlapping information. At the leader layer, they participate in a similar game, whose state dynamics are governed by a conditional mean‐field forward‐backward stochastic differential equation. By applying the maximum principle with partial information and the completion of the square technique, this paper derives the open‐loop Stackelberg–Nash equilibrium with overlapping information. It is shown that this equilibrium admits a state feedback representation, provided that an associated system of Riccati equations is solvable. As an application, this model is used to study a continuous‐time principal‐agent problem.