Abstract
This paper studies a linear quadratic nonzero sum differential game problem with asymmetric information. Compared with the existing literature, a distinct feature is that the information available to players is asymmetric. Nash equilibrium points are obtained for several classes of asymmetric information by stochastic maximum principle and technique of completion square. The systems of some Riccati equations and forward-backward stochastic filtering equations are introduced and the existence and uniqueness of the solutions are proved. Finally, the unique Nash equilibrium point for each class of asymmetric information is represented in a feedback form of the optimal filtering of the state, through the solutions of the Riccati equations.
Highlights
Throughout this article, we denote by Rk the k-dimensional Euclidean space, Rk×l the collection of k × l matrices
We introduce some Riccati equations and represent the unique Nash equilibrium point in a feedback form of the optimal filtering of the state with respect to the corresponding asymmetric information, through the solutions of the Riccati equations
We present existence and uniqueness for the solutions of the forward-backward stochastic differential equation (FBSDE, for short), whose dynamics is described by dx = b (t, x, y) dt + σ (t, x, y) dW
Summary
Throughout this article, we denote by Rk the k-dimensional Euclidean space, Rk×l the collection of k × l matrices. The problem is, under the setting of asymmetric information, to look for (u1(⋅), u2(⋅)) ∈ U1 × U2 which is called the Nash equilibrium point of the game, such that. When the available information is partial or asymmetric, we need to derive the corresponding optimal filtering of the states and adjoint variables which will be used to represent the Nash equilibrium points. It is very difficult to obtain the equations satisfied by the optimal filtering when the available information is asymmetric for Player 1 and Player 2. We introduce some Riccati equations and represent the unique Nash equilibrium point in a feedback form of the optimal filtering of the state with respect to the corresponding asymmetric information, through the solutions of the Riccati equations.
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