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In this book we discuss the existence and uniqueness of solutions of backward stochastic differential equatios (BSDE) with jumps under Lipschitzian and non-Lipschitzian conditions and with bounded and unbounded stopping times as terminal times. We also discuss the reflecting BSDE (RBSDE) with jumps and forward-backward SDE (FBSDE) with jumps, which are proved to be useful in applications of stochastic control and financial market. We research on its applications to integro-differential equations (IDE), where a new Feynman-Kac formula for the solution to IDE and the existence of the Sobolev solution to the IDE with non-Lipschitzian force are obtained. Furthermore, one kind of Hamilton-Jacobi-Bellman equation with integral term, which is proved to be important and useful in the optimal control and optimal consumption in finance, is also researched and its viscosity solution is expressed by the solution of some BSDE with jumps. Moreover, the applications to backward stochastic partial differential equations (BSPDE), to stochastic control, and to stochastic Riccati equations, all with jumps, are developed. Three Appendices: a martingale representation theorem, a generalization of Kunita-Ito's formula and the stochastic quadratic control, all for systems with jumps, are provided. Contents
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