STRONG CONVERGENCE AND STABILITY OF THE SEMI-TAMED AND TAMED EULER SCHEMES FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS UNDER NON-GLOBAL LIPSCHITZ CONDITION

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to substain the theoretical results are provided.