AI 中文总结
针对不可微且超线性增长漂移系数的随机微分方程，提出投影漂移随机化米尔斯坦方法，通过纳入漂移投影扩展方法，在特定条件下建立稳定性估计与残差界，实现L2意义下一阶强收敛，数值实验验证了收敛率与适用性。
AI 中文摘要
我们提出了一种用于具有不可微和超线性增长漂移系数的随机微分方程的投影漂移随机化米尔斯坦（PRM）方法。该方法通过将漂移投影纳入随机求积近似，将随机化米尔斯坦方法扩展到全局利普希茨设置之外。此外，与现有的用于具有超线性增长漂移系数的随机微分方程的一阶米尔斯坦型方法不同，该方法不需要漂移系数的空间可微性。在漂移的适当多项式利普希茨和单边利普希茨条件以及扩散的标准正则性假设下，我们建立了一步均方稳定性估计并导出所需的局部残差界。这些估计产生了PRM方法在L2意义下的一阶强收敛。数值实验证实了理论收敛速度，并证明了该方法对具有不可微和超线性增长漂移的随机微分方程的适用性。
英文摘要
We propose a projected drift-randomized Milstein (PRM) method for stochastic differ ential equations with non-differentiable and super-linearly growing drift coefficients. The method extends the randomized Milstein approach beyond the globally Lipschitz setting by incorporating a drift projection into the randomized quadrature approximation. Moreover, unlike existing first-order Milstein-type methods for SDEs with super-linearly growing drift coefficients, the proposed method does not require spatial differentiability of the drift coefficient. Under suitable polynomial Lipschitz and one-sided Lipschitz conditions on the drift, together with standard regularity assumptions on the diffusion, we establish a one-step mean-square stability estimate and derive the required local residual bounds. These esti mates yield first-order strong convergence of the PRM method in the L2-sense. Numerical experiments confirm the theoretical convergence rate and demonstrate the applicability of the method to SDEs with non-differentiable and super-linearly growing drifts.
