The Gaussian random walk, sampling Brownian motion, and the Riemann zeta function

We consider the Gaussian random walk (one-dimensional random walk with normally distributed increments), and in particular the moments of its maximum M. Explicit expressions for all moments of M are derived in terms of Taylor series with coefficients that involve the Riemann zeta function. We build upon the work of Chang and Peres (1997) on P(M=0) and Bateman’s formulas on Lerch’s transcendent. Our result for E(M) completes earlier work of Kingman (1965), Siegmund (1985), and Chang and Peres (1997). The maximum M shows up in a range of applications, such as sequentially testing for the drift of a Brownian motion, corrected diffusion approximations, simulation of Brownian motion, option pricing, thermodynamics of a polymer chain, and queueing systems in heavy traffic. Some of these applications are discussed, as well as several issues open for further research. This talk is based on joint work with A.J.E.M. Janssen.