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The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

Yin, Juliang and Mao, Xuerong, NSF of Guangdong Province (Funder) (2008) The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. Journal of Mathematical Analysis and Applications, 346 (2). pp. 345-358. ISSN 0022-247X

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Abstract

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.