Strong convergence of split-step theta methods for non-autonomous stochastic differential equations

Abstract

In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a onesided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments.

High‐order split‐step theta methods for non‐autonomous stochastic differential equations with non‐globally Lipschitz continuous coefficients

“…It is easy to verify that all of these SDEs (5.3)-(5.5) satisfy Assumptions 2.1, 3.1, and 4.1. The order of strong convergence for the split-step backward Euler method is 1=2 (see [5] and [13] for more details ). Therefore, we use the numerical solution computed by the split-step backward Euler method [5] with stepsize h D 2 19 as the 'exact' solution.…”

“…Wang and Gan [12] proved the strong convergence of the split-step one-leg theta method under a one-sided Lipschitz condition on the drift coefficient. Under the same conditions, Yue et al [13] proved that the split-step theta method [6] is strong convergent. There are also similar results to the split-step backward Euler method [14] and the balanced-type scheme [15].…”

“…Yue et al [13] proved that its order of strong convergence is only 1/2 under the one-sided Lipschitz condition on the drift coefficient.…”

Double-implicit and split two-step Milstein schemes for stochastic differential equations

“…Zong et al [27][28][29] utilize split-step techniques to study linear theta Euler and Milstein methods. For more on this method, we refer to [6,19,25].…”

Multi-level Monte Carlo methods with the Truncated Euler-Maruyama Scheme for Stochastic Differential Equations

“…Compared to the explicit methods mentioned above, the methods with implicit term have better convergence property in approximating non-global Lipschitz SDEs with the trade-off of the relatively expensive computational cost. We just mention a few of the works [15,29,32] and the references therein.…”