scholarly journals Stochastic Linear Quadratic Stackelberg Differential Game with Overlapping Information

2020 ◽  
Vol 26 ◽  
pp. 83
Author(s):  
Jingtao Shi ◽  
Guangchen Wang ◽  
Jie Xiong
Keyword(s):  
Differential Game ◽  
Direct Calculation ◽  
Stackelberg Equilibrium ◽  
Equilibrium Strategy ◽  
Optimal Controls ◽  
State Variables ◽  
Linear Quadratic ◽  
Stochastic Filtering ◽  
Stackelberg Differential Game ◽  
The Cost

This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term “overlapping” means that there are common part between the follower’s and the leader’s information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to give the state estimate feedback representation of the Stackelberg equilibrium strategy, while its solvability is a rather difficult open problem. A special case is then studied and is applied to the continuous-time principal-agent problem.

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 Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kai Du ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Differential System ◽  
Optimization Problems ◽  
Stackelberg Game ◽  
Mean Field ◽  
Open Loop ◽  
Stackelberg Equilibrium ◽  
Linear Quadratic ◽  
Stackelberg Differential Game ◽  
Lq Optimal Control

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

scholarly journals Differential Game Analyses of Logistics Service Supply Chain Coordination by Cost Sharing Contract

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haifeng Zhao ◽  
Bin Lin ◽  
Wanqing Mao ◽  
Yang Ye
Keyword(s):  
Supply Chain ◽  
Differential Game ◽  
Cost Sharing ◽  
Supply Chain Coordination ◽  
Coordination Mechanism ◽  
Stackelberg Equilibrium ◽  
Logistics Service ◽  
Service Supply Chain ◽  
The Cost ◽  
Relationship Of

Cooperation of all the members in a supply chain plays an important role in logistics service. The service integrator can encourage cooperation from service suppliers by sharing their cost during the service, which we assume can increase the sales by accumulating the reputation of the supply chain. A differential game model is established with the logistics service supply chain that consists of one service integrator and one supplier. And we derive the optimal solutions of the Nash equilibrium without cost sharing contract and the Stackelberg equilibrium with the integrator as the leader who partially shares the cost of the efforts of the supplier. The results make the benefits of the cost sharing contract in increasing the profits of both players as well as the whole supply chain explicit, which means that the cost sharing contract is an effective coordination mechanism in the long-term relationship of the members in a logistics service supply chain.

Linear quadratic control problems of stochastic Volterra integral equations

2018 ◽  
Vol 24 (4) ◽  
pp. 1849-1879 ◽  
Author(s):  
Tianxiao Wang
Keyword(s):  
Integral Equations ◽  
Stochastic Systems ◽  
Open Loop ◽  
Volterra Integral Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
The Cost ◽  
Quadratic Control Problems

This paper is concerned with linear quadratic control problems of stochastic differential equations (SDEs, in short) and stochastic Volterra integral equations (SVIEs, in short). Notice that for stochastic systems, the control weight in the cost functional is allowed to be indefinite. This feature is demonstrated here only by open-loop optimal controls but not limited to closed-loop optimal controls in the literature. As to linear quadratic problem of SDEs, some examples are given to point out the issues left by existing papers, and new characterizations of optimal controls are obtained in different manners. For the study of SVIEs with deterministic coefficients, a class of stochastic Fredholm−Volterra integral equations is introduced to replace conventional forward-backward SVIEs. Eventually, instead of using convex variation, we use spike variation to obtain some additional optimality conditions of linear quadratic problems for SVIEs.

SOLUTION OF A SINGULAR ZERO-SUM LINEAR-QUADRATIC DIFFERENTIAL GAME BY REGULARIZATION

2014 ◽  
Vol 16 (02) ◽  
pp. 1440007 ◽  
Author(s):  
JOSEF SHINAR ◽  
VALERY Y. GLIZER ◽  
VLADIMIR TURETSKY
Keyword(s):  
Differential Game ◽  
Perturbation Technique ◽  
Weighting Coefficient ◽  
Linear Quadratic ◽  
Control Cost ◽  
Singular Perturbation Technique ◽  
Cost Functional ◽  
Cheap Control ◽  
Zero Sum ◽  
The Cost

A linear-quadratic zero-sum singular differential game, where the cost functional does not contain the minimizer's control cost, is considered. Due to the singularity, the game cannot be solved either by applying the MinMax principle of Isaacs, or by using the Bellman–Isaacs equation method. In this paper, the solution of the singular game is obtained by using an auxiliary differential game with the same equation of dynamics and with a similar cost functional augmented by an integral of the square of the minimizer's control multiplied by a small positive weighting coefficient. This auxiliary game is a regular cheap control zero-sum differential game. For the analysis of such a cheap control differential game, in the present paper a singular perturbation technique is applied. Based on this analysis, the minimizing control sequence and the maximizer's optimal strategy in the original (singular) game are derived. Moreover, the existence of the value of the original game is established and its expression is derived. The solution is illustrated by an interception example.

scholarly journals ON A NEW PARADIGM OF OPTIMAL REINSURANCE: A STOCHASTIC STACKELBERG DIFFERENTIAL GAME BETWEEN AN INSURER AND A REINSURER

2018 ◽  
Vol 48 (02) ◽  
pp. 905-960 ◽  
Author(s):  
Lv Chen ◽  
Yang Shen
Keyword(s):  
Differential Game ◽  
Continuous Time ◽  
Stackelberg Game ◽  
General Setting ◽  
Random Coefficients ◽  
Stackelberg Equilibrium ◽  
New Paradigm ◽  
Optimal Reinsurance ◽  
Stackelberg Differential Game ◽  
Special Case

AbstractThis paper proposes a new continuous-time framework to analyze optimal reinsurance, in which an insurer and a reinsurer are two players of a stochastic Stackelberg differential game, i.e., a stochastic leader-follower differential game. This allows us to determine optimal reinsurance from joint interests of the insurer and the reinsurer, which is rarely considered in the continuous-time setting. In the Stackelberg game, the reinsurer moves first and the insurer does subsequently to achieve a Stackelberg equilibrium toward optimal reinsurance arrangement. Speaking more precisely, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. Under utility maximization criteria, we study the game problem starting from the general setting with generic utilities and random coefficients to the special case with exponential utilities and constant coefficients. In the special case, we find that the reinsurer applies the variance premium principle to calculate the optimal reinsurance premium and the insurer's optimal ceding/retained proportion of insurance risk depends not only on the risk aversion of itself but also on that of the reinsurer.

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