具有超线性漂移的随机微分方程的路径斜对称离散化
Pathwise skew-symmetric discretisation for SDEs with superlinear drift
浏览论文内容
中文总结 AI 辅助
研究具有超线性漂移的随机微分方程，提出路径斜对称离散化方法，通过特定方式耦合噪声增量用于MLMC框架，建立收敛性并分析复杂度，与驯服欧拉方案比较，数值实验验证了方法的有效性。
中文摘要 AI 辅助
斜对称离散化最近被提出作为一种新的稳健模拟方法，用于弱逼近具有非全局Lipschitz漂移的随机微分方程（SDE）。本文通过将噪声增量表示为斜正态分布并与驱动布朗增量耦合，开发了该方案的路径版本，使其能够用于多层蒙特卡罗（MLMC）框架。在适当条件下，建立了\(L^2\)中1/2阶的强收敛性。随后，相关的MLMC估计器的计算复杂度为\(\mathcal{O} \bigl(\varepsilon^{-2} (\log (1/\varepsilon))^2 \bigr)\)以实现均方误差\(\varepsilon^2\)。接着将该方案与驯服欧拉方案进行分析比较。在强内向漂移状态下，远离稳定区域时，斜对称方案中向错误方向移动的概率趋于消失，而驯服欧拉方案有非平凡概率。在MLMC设置中，使用一维随机金兹堡 - 朗道模型，指定了通过路径斜对称方案获得的耦合水平差的渐近方差低于通过驯服欧拉方案获得的渐近方差的步长范围。几个模型示例的数值实验支持了强收敛的理论速率，并证明了所得MLMC在超线性漂移设置中的稳定性和有效性。
英文摘要
The skew-symmetric discretisation has recently been proposed as a new robust simulation method for weakly approximating stochastic differential equations (SDEs) with non-globally Lipschitz drift. This work develops a pathwise version of the scheme by representing the noise increment as a skew-normal distribution and coupling it with the driving Brownian increments, thereby enabling its use in the multilevel Monte Carlo (MLMC) framework. Under suitable conditions, we establish strong convergence of order 1/2 in $L^2$. Subsequently, the associated MLMC estimator is shown to have computational complexity ${O} \bigl(\varepsilon^{-2} (\log (1/\varepsilon))^2 \bigr)$ to achieve a mean-squared error $\varepsilon^2$. We then analytically compare the proposed scheme with the tamed Euler scheme, another benchmark for robust discretisation. Under a strong inward-drift regime with the current state being far from the stable region, we show that the probability of moving in the wrong direction tends to vanish in the skew-symmetric scheme, whereas the tamed Euler scheme makes such moves with a non-trivial probability. Furthermore, in the MLMC setting, employing a one-dimensional stochastic Ginzburg-Landau model, we specify the range of step sizes for which the asymptotic variance of the coupled level difference obtained via the pathwise skew-symmetric scheme is lower than that obtained via the tamed Euler scheme. Numerical experiments on several model examples support the theoretical rate of strong convergence and demonstrate the stability and effectiveness of the resulting MLMC in the superlinear drift setting.
