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May 2022 Convergence of jump processes with stochastic intensity to Brownian motion with inert drift
Clayton Barnes
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Bernoulli 28(2): 1491-1518 (May 2022). DOI: 10.3150/21-BEJ1399

Abstract

Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.

Funding Statement

The preparation of this manuscript was partially supported by FNS 200021_175728/1.

Acknowledgements

CB is a Zuckerman Postdoctoral scholar at Technion-Israel’s Institute of Technology, Industrial Engineering and Management, Haifa, Israel, 32000. During this research the author was graduate student at the University of Washington and visited Universidad de Chile.

Citation

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Clayton Barnes. "Convergence of jump processes with stochastic intensity to Brownian motion with inert drift." Bernoulli 28 (2) 1491 - 1518, May 2022. https://doi.org/10.3150/21-BEJ1399

Information

Received: 1 October 2018; Revised: 1 February 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1399

Keywords: Brownian motion , discrete approximation , Local time , Random walk

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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