On a first hit distribution of the running maximum of Brownian motion

Julien  Randon-Furling; Paavo Salminen; Pierre Vallois

doi:10.1016/j.spa.2021.12.015

On a first hit distribution of the running maximum of Brownian motion

Julien Randon-Furling
, Paavo Salminen
, Pierre Vallois

Mathematics

Research output: Contribution to journal › Article › Scientific › peer-review

2 Citations (Scopus)

39 Downloads (Pure)

Abstract

Let (S _t) _t≥0 be the running maximum of a standard Brownian motion (B _t) _t≥0 and T _m≔inf{t;mS _t<t},m>0. In this note we calculate the joint distribution of T _m and B _{T
_m}. The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

Original language	English
Pages (from-to)	1204-1221
Number of pages	18
Journal	Stochastic Processes and their Applications
Volume	150
DOIs	https://doi.org/10.1016/j.spa.2021.12.015
Publication status	Published - Aug 2022
MoE publication type	A1 Journal article-refereed

Funding

Paavo Salminen thanks Magnus Ehrnrooth’s foundation, Finland for financial support.

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10.1016/j.spa.2021.12.015Licence: CC BY

1-s2.0-S0304414921002258-mainFinal published version, 1.72 MBLicence: CC BY

https://urn.fi/URN:NBN:fi-fe2022122973993

Cite this

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title = "On a first hit distribution of the running maximum of Brownian motion",

abstract = "Let (S t) t≥0 be the running maximum of a standard Brownian motion (B t) t≥0 and T m≔inf\{t;mS t0. In this note we calculate the joint distribution of T m and B T m . The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.",

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T1 - On a first hit distribution of the running maximum of Brownian motion

AU - Randon-Furling, Julien

AU - Salminen, Paavo

AU - Vallois, Pierre

PY - 2022/8

Y1 - 2022/8

N2 - Let (S t) t≥0 be the running maximum of a standard Brownian motion (B t) t≥0 and T m≔inf{t;mS t0. In this note we calculate the joint distribution of T m and B T m . The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

AB - Let (S t) t≥0 be the running maximum of a standard Brownian motion (B t) t≥0 and T m≔inf{t;mS t0. In this note we calculate the joint distribution of T m and B T m . The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

U2 - 10.1016/j.spa.2021.12.015

DO - 10.1016/j.spa.2021.12.015

M3 - Article

SN - 0304-4149

VL - 150

SP - 1204

EP - 1221

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

ER -

On a first hit distribution of the running maximum of Brownian motion

Abstract

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