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

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

R. A. Doney and M. S. Savov

Source: Ann. Probab. Volume 38, Number 1 (2010), 316-326.

Abstract

If X is a stable process of index α∈(0, 2) whose Lévy measure has density cx^−α−1 on (0, ∞), and S₁=sup_0<t≤1X_t, it is known that P(S₁>x)∽Aα⁻¹x^−α as x→∞ and P(S₁≤x)∽Bα⁻¹ρ⁻¹x^αρ 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)∽Bx^αρ−1 as x↓0. Similar results are obtained for related densities.

First Page: Show

Primary Subjects: 60J30, 60F15

Keywords: Stable process; stable meander; supremum; passage time density; asymptotic behavior

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1264434000
Digital Object Identifier: doi:10.1214/09-AOP479
Zentralblatt MATH identifier: 05678681
Mathematical Reviews number (MathSciNet): MR2599201

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