Convergence and Stability of Split-Step-Theta Methods for Stochastic Differential Equations With Jumps Under Non-Global Lipschitz Coefficients

Abstract

In this paper, we investigate the convergence and stability of the split step theta method (SSTM) and its compensated form for stochastic differential equations with jumps (SDEwJs) under non-global Lipschitz condition of the drift term. The methods converge strongly to the exact solution in the root mean square with order 1/2. Stability analysis reveals that the compensated split-step-theta method (CSSTM) holds the A-stability property for θ ∈ [1/2, 1] for both linear and nonlinear cases. For a linear test equation with a negative drift and positive jump coefficients, there exists θ ≤ 1/2 for which the SSTM is A-stable. This overcome the barrier of θ by D. J. Higham & P. E. Kloeden (2006) and X. Wang & S. Gan (2010). In the nonlinear case the SSTM holds the B-stability property. We give some numerical experiments to illustrate our theoretical results.