Fractional Brownian motion is a paradigmatic non-Markovian Gaussian process characterized by the Hurst exponent $H$. Despite its simple scaling form, extremal observables of fractional Brownian motion are notoriously difficult to analyze, and exact results are rare. We consider the mean of the maximum $M_H\left(1\right)={sup}_{0\le t\le 1}{B}_H\left(t\right)$ of standard fractional Brownian motion and put forward the conjecture $\langle M_H\left(1\right)\rangle =\frac{1}{H\sqrt{2\pi }},0<H\le 1$. The formula is exact for Brownian motion, $H=1/2$, and at the degenerate Gaussian endpoint $H=1$. We provide numerical evidence over a range of Hurst exponents and give a heuristic motivation based on a continuum form of the Kac-Spitzer identity. The main analytical test is obtained by comparing the conjecture with the perturbative density of the maximum derived by Delorme and Wiese for $H=1/2+\varepsilon$. After accounting for normalization conventions, the perturbative density gives $\langle M_{1/2+\varepsilon }\left(1\right)\rangle =\sqrt{\frac{2}{\pi }\phantom{{}^1}}\left(1-2\varepsilon \right)+O\left({\varepsilon }^2\right)$, in exact agreement with the first-order expansion of the conjecture. These results provide nontrivial support for a simple expression for an otherwise difficult non-Markovian extremal observable.

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• Stochastic processes