Arutkin Maxence, Walter Benjamin, Wiese Kay Jörg
UMR CNRS 7083 Gulliver, ESPCI Paris, 10 rue Vauquelin, 75005 Paris, France.
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
Phys Rev E. 2020 Aug;102(2-1):022102. doi: 10.1103/PhysRevE.102.022102.
We study the first-passage time, the distribution of the maximum, and the absorption probability of fractional Brownian motion of Hurst parameter H with both a linear and a nonlinear drift. The latter appears naturally when applying nonlinear variable transformations. Via a perturbative expansion in ɛ=H-1/2, we give the first-order corrections to the classical result for Brownian motion analytically. Using a recently introduced adaptive-bisection algorithm, which is much more efficient than the standard Davies-Harte algorithm, we test our predictions for the first-passage time on grids of effective sizes up to N_{eff}=2^{28}≈2.7×10^{8} points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.
我们研究了具有线性和非线性漂移的赫斯特参数(H)的分数布朗运动的首次通过时间、最大值分布和吸收概率。当应用非线性变量变换时，非线性漂移自然出现。通过在(\epsilon = H - \frac{1}{2})中的微扰展开，我们解析地给出了布朗运动经典结果的一阶修正。使用最近引入的自适应二分算法，其比标准的戴维斯 - 哈特算法效率高得多，我们在有效尺寸高达(N_{eff} = 2^{28} \approx 2.7×10^{8})个点的网格上测试了我们对首次通过时间的预测。理论与模拟之间的一致性非常好，并且精度远远超过仅通过标度所能获得的结果。
