On a Random Walk Between a Reflecting and an Absorbing Barrier

Abstract

We consider a Wiener process between a reflecting and an absorbingbarrier and derive a series solution for the transition density ofthe process and for the density of the time to absorption. If thedrift is towards the reflecting barrier, the variance is not toolarge, and the distance of the barriers is not too small, the leadingterm of the series derives from imaginary solutions of the basiceigenvalue equation of this problem. It is shown that these leadingterms often make the dominant contribution to the complete series.Finally, we consider previous attempts by Fürth [3], Cox and Miller[1], and by Goel and Richter-Dyn [5] to solve the stated problemand point out, in some detail, why their solutions are wrong.