The method of weighted likelihood functions

Abstract

A probabilistic way to derive first exit distributions over straight lines and curved boundaries is presented in this section. It uses the fact that mixtures of likelihood functions are positive martingales. Although the basic result of this section is well known (cf. Robbins-Siegmund (1973)), its relation to other methods for computing exit distributions was left in the dark until recently. Surprisingly the connection between the approach described here and the general method of images, described in the preceding section, is basic and simple: both methods are equivalent up to time inversion. We shall develop this connection in detail. It will lead us to a simple martingale proof of Theorem 1.1 at the end of this section.