Existence and Properties of Pseudo-Inverses for Bessel and Related Processes

Abstract

In Chapter 6, we have shown the existence of a decreasing pseudo-inverse for the martingale $(M_{t}:=\exp\left(B_{t}-\frac{t}{2}\right),t\geq0)$. We shall now explore this notion in a more general framework, starting with the case of Bessel (and some related) processes. We show in particular that the tail probabilities of a Bessel process of index ν≥1/2 increase with respect to time; in fact it is the distribution function of a random time which is related to first and last passage times of Bessel processes.