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Consider the all-time maximum of a Brownian motion with negative drift.Assume that this process is sampled at certain points in time, where the timebetween two consecutive points is rendered by an Erlang distribution with mean$1/\omega$. The family of Erlang distributions covers the range betweendeterministic and exponential distributions. We show that the averageconvergence rate as $\omega\to\infty$ for all such Erlangian sampled Brownianmotions is $O(\omega^{-1/2})$, and that the constant involved in $O$ rangesfrom $-\zeta(1/2)/\sqrt{2\pi}$ for deterministic sampling to $1/\sqrt{2}$ forexponential sampling. The basic ingredients of our analysis are a finite-seriesexpression for the expected maximum, an asymptotic expansion of$\sum_{j=1}^{k-1}(1-\exp(2\pi i j/k))^{-s}$, $s\in\mathbb{R}$, as $k\to\infty$using Euler-Maclaurin summation, and Fourier sampling of functions analytic inan open set containing the closed unit disk.

Asymptotics of the maximum of Brownian motion under Erlangian sampling