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

doi:10.1016/S0252-9602(17)30087-5

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (5): 1497-1518.doi: 10.1016/S0252-9602(17)30087-5

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

Tao HAO¹, Juan LI²

1. School of Statistics, Shandong University of Finance and Economics, Jinan 250014, China;
2. School of Mathematics and Statistics, Shandong University, Weihai 264209, China

Received:2015-05-29 Revised:2016-11-24 Online:2017-10-25 Published:2017-10-25
Contact: Juan LI,E-mail:juanli@sdu.edu.cn E-mail:juanli@sdu.edu.cn
Supported by:
The work is supported by the NSF of China (11071144, 11171187, 11222110 and 71671104), Shandong Province (BS2011SF010, JQ201202), SRF for ROCS (SEM); Program for New Century Excellent Talents in University (NCET-12-0331), 111 Project (B12023), the Ministry of Education of Humanities and Social Science Project (16YJA910003) and Incubation Group Project of Financial Statistics and Risk Management of SDUFE.

Abstract

Abstract:

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.

Key words: 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

Tao HAO, Juan LI. BSDES IN GAMES, COUPLED WITH THE VALUE FUNCTIONS. ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS[J].Acta mathematica scientia,Series B, 2017, 37(5): 1497-1518.

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

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