论文标题
与漂移的分数布朗运动的预期至上的界限
Bounds for expected supremum of fractional Brownian motion with drift
论文作者
论文摘要
我们为$ \ sup_ {t \ geqslant 0} \ {b_h（t） - t \} $的平均$ {\ mathscr m}（h）$提供上限和下限我们在（半）闭合形式中找到界限，区分（0，\ frac {1} {2}] $和$ h \ in [\ frac {1} {2}，1）$，在以前的政权中，呈现了一个数值的过程，从而大大减少了上限。对于$ 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} {2}，1）$衍生的上限和下限具有很强的相似形状。我们还为$ \ sup_ {t \ in [0,1]} b_h（t）$，$ h \ in（0，\ tfrac {1} {2}] $的平均值得出了一个新的上限，该界限在$ h = \ tfrac {1} {1} {2} $。
We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, 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 $\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\tfrac{1}{2}]$, which is tight around $H=\tfrac{1}{2}$.
