Brownian motion and a sharply curved boundary
A. D. Barbour · Advances in Applied Probability · 1981
Daniels (1974) reduced the problem of approximating the distribution of the maximum size of a closed epidemic to that of finding the distribution of max 0≦t≦2 { W ( t ) – N 1/2 c ( 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.