Ladder heights, Gaussian random walks and the Riemann zeta function

Let {S n : 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 S τ is the first ladder height. Then the expected value E θ S τ , 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.