Anticipated backward stochastic differential equations with jumps under the non-Lipschitz condition

This paper deals with a class of anticipated backward stochastic differential equations with Poisson jumps (ABSDEJs). We first show that there is a duality between anticipated backward stochastic differential equations with jumps and stochastic differential delay equations with jumps (SDDEJs). Then, we prove the existence and uniqueness of adapted solutions and Lp solutions for such ABSDEJs under the non-Lipschitz conditions as well as a comparison theorem is obtained through constructing some iterative equations which are different from iterative equations in Peng and Yang (2009).