随机混沌系统路径灵敏度的重要抽样
Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems
摘要：
提出了一种新的混沌随机微分方程路径灵敏度估计器。通过在原始和扰动的随机微分方程之间引入一个弹簧项，我们得到了一个新的估计重要抽样。与标准路径估计的指数增加相比，新估计的方差仅随时间$T,$线性增加。我们比较了我们的估计和Malliavin估计，并将它们推广到多级蒙特卡罗方法，进一步提高了计算效率。最后，我们还考虑了用这种小波动率的估计来逼近混沌常微分方程的不变测度的灵敏度。此外，Richardson-Romberg外推的波动率参数给出了一个更准确和有效的估计。数值实验支持我们的分析。
英文摘要：
This paper proposes a new pathwise sensitivity estimator for chaotic SDEs. By introducing a spring term between the original and perturbated SDEs, we derive a new estimator by importance sampling. The variance of the new estimator increases only linearly in time $T,$ compared with the exponential increase of the standard pathwise estimator. We compare our estimator with the Malliavin estimator and extend both of them to the Multilevel Monte Carlo method, which further improves the computational efficiency. Finally, we also consider using this estimator for the SDE with small volatility to approximate the sensitivities of the invariant measure of chaotic ODEs. Furthermore, Richardson-Romberg extrapolation on the volatility parameter gives a more accurate and efficient estimator. Numerical experiments support our analysis.
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