Integral-Partial Differential Equations of Isaacs' Type Related to Stochastic Differential Games with Jumps

摘要

In this paper we study zero-sum two-player stochastic differential games withjumps with the help of theory of Backward Stochastic Differential Equations(BSDEs). We generalize the results of Fleming and Souganidis [10] and those byBiswas [3] by considering a controlled stochastic system driven by ad-dimensional Brownian motion and a Poisson random measure and by associatingnonlinear cost functionals defined by controlled BSDEs. Moreover, unlike theboth papers cited above we allow the admissible control processes of bothplayers to depend on all events occurring before the beginning of the game.This quite natural extension allows the players to take into account suchearlier events, and it makes even easier to derive the dynamic programmingprinciple. The price to pay is that the cost functionals become randomvariables and so also the upper and the lower value functions of the game are apriori random fields. The use of a new method allows to prove that, in fact,the upper and the lower value functions are deterministic. On the other hand,the application of BSDE methods [18] allows to prove a dynamic programmingprinciple for the upper and the lower value functions in a verystraight-forward way, as well as the fact that they are the unique viscositysolutions of the upper and the lower integral-partial differential equations ofHamilton-Jacobi-Bellman-Isaacs' type, respectively. Finally, the existence ofthe value of the game is got in this more general setting if Isaacs' conditionholds.