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 uppervaluefunctions of SDGs, wher...
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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 uppervaluefunctions of SDGs, where the lower and the uppervaluefunctions 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 uppervaluefunctions 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 uppervaluefunctions. Using a new method, we characterize the pair(W, U) consisting of the lower and the uppervaluefunctions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.
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