On the law of terminal value of additive martingales in a remarkable branching stable process

Abstract

We give an explicit description of the law of terminal value W of additive martingales in a remarkable branching stable process. We show that the right tail probability of the terminal value decays exponentially fast and the left tail probability follows that −logP(W<x)∼12(logx)2 as x→0+. These are in sharp contrast with results in the literature such as Liu (2000, 2001) and Buraczewski (2009). We further show that the law of W is self-decomposable, and therefore, possesses a unimodal density. We specify the asymptotic behavior at 0 and at +∞ of the latter.