On the tail distributions of the supremum and the quadratic variation of a càdlàg local martingale

Abstract

We study a tail property of the distribution of the supremum and the quadratic variation of a local martingale. In the case when the local martingale is continuous, there are works by Azéma, Gundy, and Yor [1], Novikov [9], Elworthy, Li, and Yor [2], Madan and Yor [8], Takaoka [10] etc. Recently, Liptser and Novikov [7] extended these studies to the case of a local martingale with uniformly bounded jumps; here is their main result: Theorem 1.1 Let M = be a locally square integrable càdlàg martingale defined on a filtered probability space with standard general conditions. Assume that 〈M〉∞ = lim t→∞ 〈M〉t < ∞ a.s and is uniformly integrable, where τ is the set of stopping times τ. Then (i) 0 ≤ E[M ∞] ≤ E[M ∞+ ] K and and for K > 0 and > 0, then λ} = limλ→∞ λP{ 〈M〉∞ > λ} = 2/πE[M∞]. © 2008 Springer-Verlag Berlin Heidelberg.