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We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables y, z and an only square integrable terminal data. Let L N be the Lipschitz constant of the coefficient on the ball B(0, N ) of R d × R dr . We prove that if L N = O( √ log N ), then the corresponding BSDE has a unique solution. Moreover, the stability of the solution is established under the same assumptions. In the case where the terminal data is bounded, we establish the existence and uniqueness of the solution also when the coefficient has an arbitrary growth (in y) and without restriction on the behaviour of the Lipschitz constant L N .

We deal with backward stochastic differential equations (BSDE for short) driven by Teugel's martingales and an independent Brownian motion. We study the existence, uniqueness and comparison of solutions for these equations under a Lipschitz as well as a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove that if the Lipschitz constant LN behaves as log(N) in the ball B(0,N), then the corresponding BSDE has a unique solution which depends continuously on the on the coefficient and the terminal data. This is done with an unbounded terminal data. As application, we give a probabilistic interpretation for a large class of partial differential integral equations (PDIE for short).