The asymptotic behavior of densities related to the supremum of a stable process

Doney, R.A. and Savov, M.S. (2010) The asymptotic behavior of densities related to the supremum of a stable process. Annals of Probability, 38 (1). pp. 316-326. ISSN 2168-894X doi: 10.1214/09-AOP479

Abstract/Summary

If X is a stable process of index α∈(0, 2) whose Lévy measure has density cx−α−1 on (0, ∞), and S1=sup0<t≤1Xt, it is known that P(S1>x)∽Aα−1x−α as x→∞ and P(S1≤x)∽Bα−1ρ−1xαρ as x↓0. [Here ρ=P(X1>0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x)∽Ax−(α+1) as x→∞ and m(x)∽Bxαρ−1 as x↓0. Similar results are obtained for related densities.