Ladder heights, Gaussian random walks and the Riemann zeta function

Abstract

Let {Sn: n ≥ 0} be a random walk having normally distributed increments with mean θ and variance 1, and let τ be the time at which the random walk first takes a positive value, so that Sτ is the first ladder height. Then the expected value EθSτ, originally defined for positive θ, may be extended to be an analytic function of the complex variable θ throughout the entire complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about θ = 0 may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series; makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot.