In this paper, the stochastic theta (ST) method is investigated for a class of stochastic differential equations driven by a time-changed Brownian motion, whose coefficients are time-space-dependent and satisfy the local Lipschitz condition. It is proved that under the local Lipschitz and some additional assumptions, the ST method with $\theta\in[1/2,1]$ is strongly convergent. It is also obtained that, for all positive stepsizes, the ST method with $\theta\in[1/2,1]$ is asymptotically mean square stable under a coercivity condition. With some restrictions on the stepsize, the ST method with $\theta\in[0,1/2)$ is asymptotically mean square stable under a stronger assumption. Some numerical simulations are presented to illustrate the theoretical results.

翻译：本文研究了一类由时变布朗运动驱动的随机微分方程的随机theta（ST）方法，其系数具有时空依赖性且满足局部Lipschitz条件。证明了在局部Lipschitz条件及若干附加假设下，当$\theta\in[1/2,1]$时ST方法具有强收敛性。同时得到，在强制条件下，对于任意正步长，$\theta\in[1/2,1]$的ST方法均具有渐近均方稳定性。在步长受限的情况下，当$\theta\in[0,1/2)$时，在更强的假设下ST方法也具有渐近均方稳定性。文中通过数值模拟验证了理论结果。