We consider the Gaussian random walk (one-dimensional random walk with normally distributed increments) with negative drift, and in particular the moments of its maximum M.
We derive explicit expressions for all moments of M in terms of Taylor series about "drift=0" 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) extends Kingman's (1965) first order approximation for a drift close to zero.
The key idea in obtaining the Taylor series for the k-th moment is to differentiate its Spitzer-type expression (involving the Gaussian distribution) k + 1 times, rewrite the resulting expression in terms of Lerch's transcendent, and integrate k + 1 times. The major issue then is to determine the k + 1 integration constants, for which we invoke Euler−Maclaurin summation, among other things.
We further present sharp bounds on P(M=0) and the first two moments of M. The bounds on P(M=0) complement Siegmund's diffusion correction for the Gaussian family. We also show how our results might find important applications, particularly for queues in heavy traffic and for the equidistant sampling of Brownian motion. Indeed, 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 quality & efficiency driven regimes for large-scale service systems (e.g. call centers).