Arrow Sufficient Conditions for Optimality of Fully Coupled Forward–Backward Stochastic Differential Equations with Applications to Finance

2014 ◽  
Vol 165 (2) ◽  
pp. 639-656 ◽  
Author(s):  
Guangchen Wang ◽  
Hua Xiao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Sufficient Conditions For Optimality ◽  
Fully Coupled

scholarly journals Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hui Min ◽  
Ying Peng ◽  
Yongli Qin
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Backward Stochastic Differential Equations ◽  
Maximum Principles ◽  
Control Problems ◽  
Linear Quadratic Optimal Control ◽  
Fully Coupled

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

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.

scholarly journals Backward stochastic differential equations with unbounded generators

2019 ◽  
Vol 19 (01) ◽  
pp. 1950008 ◽  
Author(s):  
Bujar Gashi ◽  
Jiajie Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Comparison Theorem ◽  
Linear Growth ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Type Condition ◽  
Solution Pair ◽  
Picard Iterations

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

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

2018 ◽  
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 differential games where the players' state dynamics solve backward stochastic differential equations (BSDEs) 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.

Sign in / Sign up

Export Citation Format

Share Document