Electronic Communications in Probability

The tail of the maximum of Brownian motion minus a parabola

Piet Groeneboom and Nico Temme

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Abstract

We analyze the tail behavior of the maximum $N$ of $\{W(t)-t^2:t\ge0\}$, where $W$ is standard Brownian motion on $[0,\infty)$, and give an asymptotic expansion for ${\mathbb P}\{N\ge x\}$, as $x\to\infty$. This extends a first order result on the tail behavior, which can be deduced from Hüsler and Piterbarg (1999). We also point out the relation between certain results in Janson et al. (2010) and Groeneboom (2010).

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 41, 458-466.

Dates
Accepted: 24 August 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261997

Digital Object Identifier
doi:10.1214/ECP.v16-1645

Mathematical Reviews number (MathSciNet)
MR2831084

Zentralblatt MATH identifier
1244.60052

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J75: Jump processes

Keywords
Brownian motion parabolic drift maximum Airy functions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Groeneboom, Piet; Temme, Nico. The tail of the maximum of Brownian motion minus a parabola. Electron. Commun. Probab. 16 (2011), paper no. 41, 458--466. doi:10.1214/ECP.v16-1645. https://projecteuclid.org/euclid.ecp/1465261997


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References

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