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 $\varepsilon =H-1/2$, 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 $N_{\mathrm{eff}}=2^{28}\approx 2.7\times {10}^8$ points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.

中文翻译：

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分数布朗运动随漂移的极端事件：理论和数值验证。

我们研究了赫斯特参数的第一次通过时间，最大值的分布以及分数布朗运动的吸收概率 $H$具有线性和非线性漂移。当应用非线性变量变换时，后者自然出现。通过摄动扩张$\varepsilon =H-\mathrm{1个}/2$，我们对布朗运动的经典结果进行一阶校正。使用最近引入的自适应对分算法，该算法比标准的Davies-Harte算法效率更高，我们在有效尺寸最大为0的网格上测试了首次通过时间的预测${ñ}_{效}=2^{28}\approx 2.7\times {10}^8$点。理论与仿真之间的一致性非常好，并且精度远远超过仅通过缩放即可获得的精度。