phenomena, it is natural to model them with equations which involves the deterministic part and the random part including jump. The SDEs with jumps is the generalization of both deterministic part and random part with jumps. SDEs with jumps have probability theory and stochastic process as prerequisites. We refer to [@Oksa], [@Oks], [@Phi] for general notions in probability theory and stochastic process.

In this thesis, under global Lipschitz condition, we prove the existence and uniqueness of solution of SDEs with jumps. We focus on the strong convergence of the compensated stochastic theta methods (CSTM) of these equations under global Lipschitz condition. In particular, we prove that CSTM have strong convergence of order $0.5$. We investigate the stability of both CSTM and stochastic theta method (STM). For the linear case, we prove that under the assumption $\dfrac{1}{2}\leq \theta\leq 1$, CSTM holds the A-stability property. For the general nonlinear problem, we study the stability for $\theta=1$. In this case, when the drift coefficient have a negative one-sided Lipschitz coefficient, the diffusion coefficient and the jump coefficient satisfy the global Lipschitz condition, we prove that STM reproduce stability under certain step-size and the CSTM is stable for any step-size.

Most phenomena are modelised 