On the exact distributions of the maximum of the asymmetric telegraph process

Fabrizio Cinque and Enzo Orsingher

Stochastic Processes and their Applications, 2021, vol. 142, issue C, 601-633

Abstract: In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval [0,t] under the conditions that the initial velocity V(0) is either c1 or −c2 and the number of changes of direction is odd or even. For the case V(0)=−c2 the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on c1 and c2, but not on the current time t. We obtain also the unconditional distribution of the maximum for either V(0)=c1 or V(0)=−c2 and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler–Poisson–Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper.

Keywords: Telegraph process; Stochastic motions with drift; Induction principle; Bessel functions; Galilean transformations; Euler–Poisson–Darboux equations (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414921001538
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:142:y:2021:i:c:p:601-633

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2021.09.011

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().