具有非全局利普希茨系数的随机微分方程的自适应时间步长欧拉-玛丽乌玛格式：一致收敛性、稳定性与遍历性
Adaptive Time-Stepping Euler--Maruyama Scheme for SDEs with Non-Globally Lipschitz Coefficients: Uniform Convergence, Stability and Ergodicity
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中文总结 AI 辅助
本文针对非全局利普希茨系数SDE提出自适应时间步长欧拉-玛丽乌玛格式，证明其1/2阶强收敛性、矩指数稳定性与多项式遍历性，数值实验验证其精度与计算性能优于现有方法。
中文摘要 AI 辅助
本文针对漂移项与扩散项系数为非全局利普希茨型的随机微分方程（SDE），提出一种自适应时间步长欧拉-玛丽乌玛（Euler--Maruyama）格式。该格式通过在每一次迭代中动态调整时间步长，有效避免了数值不稳定性。我们证明了数值解的矩有界性，并在有限时间区间及时间一致情形下，建立了1/2阶强收敛速率。此外，该格式忠实地继承了 underlying SDE的p阶矩指数稳定性。对于长时间遍历动力学，我们证明了数值不变测度的多项式遍历性，且该数值不变测度在L^q-瓦瑟斯坦距离下，以最优的1/2阶速率收敛到underlying SDE的不变测度。数值实验验证了我们的理论结果，并表明该格式相较于多种固定步长方法及现有自适应方法，具有更优的精度与计算性能。
英文摘要
This paper develops an adaptive time-stepping Euler--Maruyama scheme for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By dynamically adjusting the timestep at each iteration, the proposed scheme effectively prevents numerical instability. We prove the moment boundedness of the numerical solution and establish a $1/2$-order strong convergence rate both on finite-time intervals and uniformly in time. Furthermore, the scheme faithfully inherits the $p$th moment exponential stability of the underlying SDE. For long-time ergodic dynamics, we establish the polynomial ergodicity of the numerical invariant measure. Moreover, we show that the numerical invariant measure converges to the invariant measure of the underlying SDE at an optimal rate of $1/2$ in the $L^q$-Wasserstein distance. Numerical experiments confirm our theoretical results and indicate the superior accuracy and computational performance of the proposed scheme over several fixed-step and existing adaptive methods.
