On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero‐sum linear quadratic stochastic differential game

2019 ◽  
Vol 41 (2) ◽  
pp. 640-667 ◽  
Author(s):  
V. Dragan ◽  
S. Aberkane ◽  
T. Morozan
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Stochastic Differential Game ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Stabilizing Solution ◽  
Zero Sum

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

scholarly journals Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fu Zhang ◽  
QingXin Meng ◽  
MaoNing Tang
Keyword(s):  
Differential Game ◽  
Stochastic Systems ◽  
Partial Information ◽  
Backward Stochastic Differential Equation ◽  
Game Problem ◽  
Stochastic Differential Game ◽  
Linear Quadratic ◽  
Independent Brownian Motion ◽  
Teugels Martingales ◽  
Zero Sum

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

A Stochastic Differential Game with Safe and Risky Choices

1988 ◽  
Vol 2 (1) ◽  
pp. 31-39
Author(s):  
J. M. McNamara
Keyword(s):  
Differential Equation ◽  
Diffusion Coefficient ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
The State ◽  
Stochastic Differential Game ◽  
Final Time ◽  
One Dimensional ◽  
Zero Sum ◽  
Drift Coefficient

This paper considers a two-person zero-sum stochastic differential game. The dynamics of the game are given by a one-dimensional stochastic differential equation whose diffusion coefficient may be controlled by the players. The drift coefficient is held constant and cannot be controlled. Player l's objective is to maximize the probability that the state at final time, T, is positive, while Player 2's objective is to maximize the probability that the state is negative.

scholarly journals An Analytic and Numerical Investigation of a Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 66
Author(s):  
Aviv Gibali ◽  
Oleg Kelis
Keyword(s):  
Differential Game ◽  
Equation Approach ◽  
Linear Quadratic ◽  
Control Cost ◽  
Dual Representation ◽  
Cost Functional ◽  
Variational Derivatives ◽  
Zero Sum ◽  
The Cost ◽  
Derivatives Of

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

scholarly journals Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 132
Author(s):  
Valery Y. Glizer
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Quadratic Functional ◽  
Linear Quadratic ◽  
Feedback Controls ◽  
Cheap Control ◽  
Optimal Values ◽  
Definition Of ◽  
Zero Sum ◽  
Equilibrium Sequence

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

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