The H-derivative of the expected supremum of fractional Brownian motion \(\{B_H(t),t\in {\mathbb {R}}_+\}\) with drift \(a\in {\mathbb {R}}\) over time interval [0, T]

$$\begin{aligned} \frac{\partial }{\partial H} {\mathbb {E}}\Big (\sup _{t\in [0,T]} B_H(t) - at\Big ) \end{aligned}$$

at \(H=1\) is found. This formula depends on the quantity \({\mathscr {I}}\), which has a probabilistic form. The numerical value of \({\mathscr {I}}\) is unknown; however, Monte Carlo experiments suggest \({\mathscr {I}}\approx 0.95\). As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as \(H\uparrow 1\).

中文翻译：

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$$H=1$$H = 1 处分数布朗运动的预期上式的导数

分数布朗运动\(\{B_H(t),t\in {\mathbb {R}}_+\}\)与漂移\ (a\in {\mathbb {R} }\)在时间间隔 [0,  T ]

$$\begin{aligned} \frac{\partial }{\partial H} {\mathbb {E}}\Big (\sup _{t\in [0,T]} B_H(t) - at\Big ) \end{对齐}$$

在\(H=1\)处找到。这个公式取决于数量\({\mathscr {I}}\)，它具有概率形式。\({\mathscr {I}}\)的数值未知；然而，蒙特卡洛实验表明\({\mathscr {I}}\approx 0.95\)。作为副产品，我们在C [0, 1] 中为分数布朗桥建立了一个弱极限定理，如\(H\uparrow 1\)。