Adv. Appl. Prob. 43, 264–275 (2011) Printed in Northern Ireland © Applied Probability Trust 2011 AN APPROXIMATION FOR THE INVERSE FIRST PASSAGE TIME PROBLEM JING-SHENG SONG, ∗ and PAUL ZIPKIN, ∗ ∗∗ Duke University Abstract We propose an approximation for the inverse first passage time problem. It is similar in spirit and method to the tangent approximation for the original first passage time problem. We provide evidence that the technique is quite accurate in many cases. We also identify some cases where the approximation performs poorly. Keywords: Inverse first passage time; Wiener process; approximation 2010 Mathematics Subject Classification: Primary 60J65 Secondary 65N21; 58J35 1. Introduction The first passage time of a stochastic process W ={W(t) : t ≥ 0} to a boundary b = b(t) is τ = inf {t> 0 : W(t) ≥ b(t)}. The first passage time (FPT) problem is to find the distribution F of τ , given b. Here we are mainly concerned with the inverse first passage time (IFPT) problem: given F , find the corresponding b. We focus on the case where W is a standard Wiener process starting at W(0) = 0. We assume that F is continuous, with a continuous, strictly positive density f . The IFPT problem has applications in biology, finance, and other areas. Recently, Song and Zipkin [12] formulated an inventory-control problem which, in certain cases, turns out to be equivalent to the IFPT problem. See the other references below for further discussion of applications and pointers to the literature on them. The IFPT problem has received considerable attention recently. Cheng et al. [2] showed that the problem is well posed—given F , there exists a unique solution (in the weak or viscosity sense). Zucca and Sacerdote [14] described two solution methods, one employing a piecewise- linear approximation of the boundary, and the other based on discretizing a Volterra integral equation. Song and Zipkin [12] presented a different algorithmic approach. Other work on this and related problems include [1], [7], and [10]. We propose a simple approximation for the IFPT problem. It is similar in spirit and method to the tangent approximation for the original FPT problem. We provide evidence that the technique is quite accurate in many cases, with errors of just a few percent. Of course, it is no substitute for the exact methods mentioned above. We also identify some problematic cases, where the approximation performs poorly. These have strongly bimodal or oscillating densities f . We suspect that such cases may prove challenging for other methods for both the FPT and the IFPT problems. Received 11 May 2010; revision received 14 October 2010. ∗ Postal address: Fuqua School of Business, Duke University, Durham, NC, USA. ∗∗ Email address: paul.zipkin@duke.edu 264