We provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$ , with $B_H(\!\cdot\!)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$ . We find bounds in (semi-) closed form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$ , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$ , the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$ , $H\in(0,\frac{1}{2}]$ , which is tight around $H=\frac{1}{2}$ .

中文翻译：

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带漂移的分数布朗运动的预期上界

我们提供平均值的上限和下限$\mathscr{M}(H)$的$\sup_{t\geq 0} \{B_H(t) - t\}$， 和$B_H(\!\cdot\!)$具有赫斯特参数的分数布朗运动的零均值、方差归一化版本$H\in(0,1)$. 我们发现（半）封闭形式的界限，区分$H\in(0,\frac{1}{2}]$和$H\in[\frac{1}{2},1)$，在前一种情况下，提出了一个数值程序，该程序大大降低了上限。为了$H\in(0,\frac{1}{2}]$，上限和下限之间的比率是有界的，而对于$H\in[\frac{1}{2},1)$派生的上限和下限具有非常相似的形状。我们还推导出平均值的新上界$\sup_{t\in[0,1]} B_H(t)$,$H\in(0,\frac{1}{2}]$, 周围很紧$H=\frac{1}{2}$.