The Solution of a Statistical Decision Problem with Moving Boundaries by the Image Method

Abstract

A point executes Brownian motion in one dimension, starting from a known initial position. Its velocity at any time is a superposition of a constant “drift” speed and a random, diffusive component, the motion taking place between two absorbing boundaries which move towards one another with constant speeds. The probability distribution for the particle position at any time is found by the method of images. The problem occurred originally in the detection of a small d.c. signal in a large amount of “white” noise, and this application is discussed.