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Nash equilibrium payoffs for stochastic differential games with jumps and coupled nonlinear cost functionals

In this paper we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games whose cost functionals are defined by a system of coupled backward stochastic differential equations. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs. For this end the problem is described equivalently by a stochastic differential game with jumps. But, however, unlike the work by Buckdahn, Hu and Li [9], here the important tool of a dynamic programming principle for stopping times has to be developed. Moreover, we prove that the lower and upper value functions are the viscosity solutions of the associated coupled systems of PDEs of Isaacs type, respectively. Our results generalize those by Buckdahn, Cardaliaguet and Rainer [7] and by Lin [16].