Open Access
2025 A note on the range of stochastic processes
Maher Boudabra, Binghao Wu
Author Affiliations +
Electron. Commun. Probab. 30: 1-3 (2025). DOI: 10.1214/24-ECP653

Abstract

A Brownian motion with drift is simply a process Vtη of the form Vtη=Bt+ηt where Bt is a standard Brownian motion and η>0. In [7], the authors showed that the underlying range Rt(Vη)=sup0stVtηinf0stVtη is equivalent to ηt a.e in the long run, i.e

Rt(Vη)tta.eη.(0.1)

In this paper, we show that (0.1) follows from a deterministic property. More precisely, we show that the long run behavior of the range of a (deterministic) function is obtainable straightaway from that of the function itself.

Citation

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Maher Boudabra. Binghao Wu. "A note on the range of stochastic processes." Electron. Commun. Probab. 30 1 - 3, 2025. https://doi.org/10.1214/24-ECP653

Information

Received: 11 May 2024; Accepted: 30 December 2024; Published: 2025
First available in Project Euclid: 13 January 2025

Digital Object Identifier: 10.1214/24-ECP653

Subjects:
Primary: 60G17 , 60J65

Keywords: Brownian motion with drift , Stochastic processes

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