Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

2020 ◽  
Vol 15 (6) ◽  
pp. 1307-1326
Author(s):  
Qingfeng Zhu ◽  
Lijiao Su ◽  
Fuguo Liu ◽  
Yufeng Shi ◽  
Yong’ao Shen ◽  
...  
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Mean Field ◽  
Stochastic Differential Games ◽  
Field Type ◽  
Doubly Stochastic

scholarly journals Mean-field backward–forward stochastic differential equations and nonzero sum stochastic differential games

2020 ◽  
pp. 2150036
Author(s):  
Yinggu Chen ◽  
Boualem Djehiche ◽  
Said Hamadène
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Mean Field ◽  
Stochastic Differential Games ◽  
Open Loop ◽  
Random Coefficients ◽  
Linear Quadratic ◽  
Uniqueness Results ◽  
Weak Monotonicity

We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.

scholarly journals Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations

Games ◽  
2018 ◽  
Vol 9 (4) ◽  
pp. 88 ◽  
Author(s):  
Alexander Aurell
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Social Cost ◽  
Price Of Anarchy ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Regularity Conditions ◽  
Field Type ◽  
Cost Functionals

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

2015 ◽  
Vol 47 (02) ◽  
pp. 355-377
Author(s):  
Qian Lin
Keyword(s):  
Nash Equilibrium ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Differential Games ◽  
Cost Functionals ◽  
Reflecting Barriers

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

scholarly journals Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

2020 ◽  
Vol 5 (2) ◽  
pp. 205-216
Author(s):  
Mostapha Abdelouahab Saouli
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Mean Field ◽  
Existence Of Solution ◽  
Maximal Solution ◽  
Existence Of A Solution ◽  
Doubly Stochastic

AbstractIn this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

scholarly journals Nash equilibria for nonzero-sum ergodic stochastic differential games

2017 ◽  
Vol 54 (4) ◽  
pp. 977-994 ◽  
Author(s):  
Samuel N. Cohen ◽  
Victor Fedyashov
Keyword(s):  
Nash Equilibrium ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Nash Equilibria ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Differential Games ◽  
Ergodic Behaviour

Abstract We consider nonzero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of ergodic backward stochastic differential equations, and prove the existence of a Nash equilibrium under generalised Isaac's conditions. We also study the case of interacting players of different type.

scholarly journals Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Mean Field ◽  
Partial Differential ◽  
Existence And Uniqueness Results ◽  
Doubly Stochastic ◽  
Uniqueness Results

Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

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