The Annals of Applied Probability

Change-point detection for Lévy processes

José E. Figueroa-López and Sveinn Ólafsson

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Abstract

Since the work of Page in the 1950s, the problem of detecting an abrupt change in the distribution of stochastic processes has received a great deal of attention. In particular, a deep connection has been established between Lorden’s minimax approach to change-point detection and the widely used CUSUM procedure, first for discrete-time processes, and subsequently for some of their continuous-time counterparts. However, results for processes with jumps are still scarce, while the practical importance of such processes has escalated since the turn of the century. In this work, we consider the problem of detecting a change in the distribution of continuous-time processes with independent and stationary increments, that is, Lévy processes, and our main result shows that CUSUM is indeed optimal in Lorden’s sense. This is the most natural continuous-time analogue of the seminal work of Moustakides [Ann. Statist. 14 (1986) 1379–1387] for sequentially observed random variables that are assumed to be i.i.d. before and after the change-point. From a practical perspective, the approach we adopt is appealing as it consists in approximating the continuous-time problem by a suitable sequence of change-point problems with equispaced sampling points, and for which a CUSUM procedure is shown to be optimal.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 717-738.

Dates
Received: November 2016
Revised: June 2017
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298928

Digital Object Identifier
doi:10.1214/17-AAP1368

Mathematical Reviews number (MathSciNet)
MR3910015

Zentralblatt MATH identifier
07047436

Subjects
Primary: 62L10: Sequential analysis 60G51: Processes with independent increments; Lévy processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62C20: Minimax procedures

Keywords
Change-point sequential detection optimal stopping CUSUM Lévy processes

Citation

Figueroa-López, José E.; Ólafsson, Sveinn. Change-point detection for Lévy processes. Ann. Appl. Probab. 29 (2019), no. 2, 717--738. doi:10.1214/17-AAP1368. https://projecteuclid.org/euclid.aoap/1548298928


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