Compensated stochastic theta methods for stochastic differential equations with jumps

In this paper, a family of compensated stochastic theta methods (CSTM), as opposed to stochastic theta methods (STM) are proposed after the introduction of a compensated Poisson process. These methods are justified to have a strong convergence order of 1/2. Further we investigate mean-square stability of the proposed methods. For a linear test equation, we show that an extension of the deterministic A-stability property holds for CSTM, if and only if 1/2⩽θ⩽1. For a general nonlinear problem, of which the drift term f has a negative one-sided Lipschitz constant and the diffusion terms g,h satisfy global Lipschitz condition, we find that backward Euler method (STM with θ=1) preserves stability under a stepsize constraint, while compensated backward Euler method (CSTM with θ=1) gives a generalization of the deterministic B-stability. Those stability results indicate that CSTM achieve superiority over STM in terms of stability.