Brownian motion and a sharply curved boundary

Abstract

Daniels (1974) reduced the problem of approximating the distribution of the maximum size of a closed epidemic to that of finding the distribution of max0≦t≦2 {W(t) – N1/2c(t)}, where c is a smooth function with a unique minimum of 0 at t = 1, and he derived an approximation to this distribution which he showed to be accurate to order N–1/4. In this paper, his approximation is shown to be accurate to order N–1/3, and a refined approximation is given which is accurate to order N–1/2 log N. The new approximation is still normal, and its accuracy is similar to that of the original approximation of a discrete process by the Wiener process.