  In this work we consider a stochastic differential equation (SDEs) with jump.
We prove the existence and the uniqueness of solution of this equation in the
strong sense under global Lipschitz condition. Generally, exact solutions of
SDEs are unknowns. The challenge is to approach them numerically. There exist
several numerical techniques. In this thesis, we present the compensated
stochastic theta method (CSTM) which is already developed in the literature. We
prove that under global Lipschitz condition, the CSTM converges strongly with
standard order 0.5. We also investigated the stability behaviour of both CSTM
and stochastic theta method (STM). Inspired by the tamed Euler scheme developed
in [8], we propose a new scheme for SDEs with jumps called compensated tamed
Euler scheme. We prove that under non-global Lipschitz condition the
compensated tamed Euler scheme converges strongly with standard order 0.5.
Inspired by [11], we propose the semi-tamed Euler for SDEs with jumps under
non-global Lipschitz condition and prove its strong convergence of order 0.5.
This latter result is helpful to prove the strong convergence of the tamed
Euler scheme. We analyse the stability behaviours of both tamed and semi-tamed
Euler scheme We present also some numerical experiments to illustrate our
theoretical results.
