TY - BOOK
T1 - Asymptotics of the maximum of Brownian motion under Erlangian sampling
AU - Janssen, A.J.E.M.
AU - Leeuwaarden, van, J.S.H.
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
M3 - Report
T3 - arXiv.org
BT - Asymptotics of the maximum of Brownian motion under Erlangian sampling
PB - s.n.
ER -