We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches is illustrated by considering some well-known SDE models.

我们引入了一个驯服的指数时间积分器，它利用随机微分方程 (SDE) 的漂移和扩散中的线性项，具有单边全局 Lipschitz 漂移项。证明了所提出方案的强收敛性，利用几何布朗运动（GBM）的有界性，我们为线性扩散项建立了 1 阶收敛性。在我们的实现中，我们说明了与现有固定步长方法相比所提出方案的效率，并将其用于自适应时间步长方案。此外，我们将该方法扩展到非线性扩散项，并表明它仍然具有竞争力。通过考虑一些众所周知的 SDE 模型来说明这些基于 GBM 的方法的效率。