Asymptotics of the maximum of Brownian motion under Erlangian sampling

TL;DR

This paper analyzes the asymptotic behavior of the maximum of a Brownian motion with negative drift when sampled at Erlang-distributed times, revealing an $O( ext{omega}^{-1/2})$ convergence rate and how it varies with sampling distribution.

Contribution

It provides a novel asymptotic analysis of the maximum of sampled Brownian motion under Erlang sampling, including explicit convergence rates and constants.

Findings

Convergence rate of $O( ext{omega}^{-1/2})$ as sampling frequency increases.

Explicit constants depending on the sampling distribution (deterministic vs exponential).

Finite-series expression and asymptotic expansion techniques used in analysis.

Abstract

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 time between two consecutive points is rendered by an Erlang distribution with mean $1/\omega$. The family of Erlang distributions covers the range between deterministic and exponential distributions. We show that the average convergence rate as $\omega\to\infty$ for all such Erlangian sampled Brownian motions is $O(\omega^{-1/2})$, and that the constant involved in $O$ ranges from $-\zeta(1/2)/\sqrt{2\pi}$ for deterministic sampling to $1/\sqrt{2}$ for exponential sampling. The basic ingredients of our analysis are a finite-series expression 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 in an open set containing the closed unit disk.