论文标题
欧拉方法的强误差分析具有分数噪声过度阻尼的广义朗格文鸟方程：非线性情况
Strong error analysis of Euler methods for overdamped generalized Langevin equations with fractional noise: Nonlinear case
论文作者
论文摘要
本文考虑了Euler和快速EULER方法的强误差分析，用于非线性抑制由分数噪声驱动的广义朗格文方程。主要难度在于处理分数布朗运动与奇异核之间的相互作用，该核通过malliavin conculus克服，以及对几个多个奇异积分的精细估计。因此，这两种方法被证明是强烈收敛的，几乎$ \ min \ {2（h+α-1），α\} $，其中$ h \ in（1/2,1）$和$α\ in（1/2,1）$和$α\ in（1-h，1）$分别表征了分数noises noises Noises Noises Noises和Sightular Kernels in deflosemular Kernelels中的奇异性水平。该结果改善了非线性案例的现有收敛顺序$ H+α-1$，并对[4]中提出的开放问题给出了积极的答案。作为理论发现的应用，我们进一步研究了基于快速Euler方法的多级蒙特卡洛模拟的复杂性，在计算所考虑方程的功能时，它的表现比标准蒙特卡洛模拟的表现更好。
This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly $\min\{2(H+α-1), α\}$, where $H \in (1/2,1)$ and $α\in(1-H,1)$ respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order $H+α-1$ of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in [4]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered equation.
