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This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715–2750, 2011 ), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs’ type combined with an algebraic equation, respectively. Furthermore, for a special case (when $$\sigma $$ σ and $$h$$ h do not depend on $$(y, z, k)$$ ( y , z , k ) ), under the Isaacs’ condition, we get the existence of the value of the game.
Applied Mathematics and Optimization – Springer Journals
Published: Jun 1, 2015
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