Dȩbicki, K, Ji, L orcid.org/0000-0002-7790-7765 and Rolski, T (2020) Exact asymptotics of component-wise extrema of two-dimensional Brownian motion. Extremes, 23 (4). pp. 569-602. ISSN 1386-1999
Abstract
We derive the exact asymptotics of
$ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty , $
where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Springer Science+Business Media, LLC, part of Springer Nature 2020. This is an author produced version of an article published in Extremes. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Component-wise extrema; Exact asymptotics; Generalised Pickands-Piterbarg constants; Quadratic programming problem; Two-dimensional Brownian motion |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 23 Jul 2020 15:27 |
Last Modified: | 17 Jun 2022 10:08 |
Status: | Published |
Publisher: | Springer Verlag |
Identification Number: | 10.1007/s10687-020-00387-y |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:163359 |