Pickands' constant at first order in an expansion around Brownian motion

In the theory of extreme values of Gaussian processes, many results are expressed in terms of the Pickands constant ${\mathcal{H}}_{\alpha }$. This constant depends on the local self-similarity exponent $\alpha$ of the process, i.e. locally it is a fractional Brownian motion (fBm) of Hurst index $H=\alpha \mathrm{/}2$. Despite its importance, only two values of the Pickands constant are known: ${\mathcal{H}}_1=1$ and ${\mathcal{H}}_2=1\mathrm{/}\sqrt{\pi }$. Here, we extend the recent perturbative approach to fBm to include drift terms. This allows us to investigate the Pickands constant ${\mathcal{H}}_{\alpha }$ around standard Brownian motion ($\alpha =1$) and to derive the new exact result $$\mathcal{H}${\alpha}=1 - (\alpha-1) \gamma{\rm E} + $\mathcal{O}$!\left( \alpha-1\right)^{2}$.