Abstract: The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non-Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi-linear elliptic type integro-differential equations is obtained.

Key words: backward stochastic differential equations(BSDEs)with jumps, unbounded stopping time, adapted solutions, convergence of solutions, quasi-linear elliptic equations, integro-differential operators