BSDES IN GAMES, COUPLED WITH THE VALUE FUNCTIONS. ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS

作者：Hao, Tao; Li, Juan^*

来源：Acta Mathematica Scientia, 2017, 37(5): 1497-1518.

DOI：10.1016/S0252-9602(17)30087-5

• McKean-Vlasov SDE
• BSDE coupled with the lower and the upper value functions
• dynamic programming principle
• mean-field BSDE
• viscosity solution
• coupled nonlocal HJB-Isaacs equation
• Isaacs' condition

We establish a new type of backward stochastic differential equations (BSDEs) connected with stochastic differential games (SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs (HJB-Isaacs) equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair (W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs' condition.

• 出版日期2017-9
• 单位山东财经大学; 山东大学