Extensions of Bougerol’s identity in law and the associated anticipative path transformations

Yuu Hariya

Stochastic Processes and their Applications, 2022, vol. 146, issue C, 311-334

Abstract: Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of the geometric Brownian motion eBt,t≥0. Bougerol’s celebrated identity in law (1983) asserts that, if β={β(t)}t≥0 is another Brownian motion independent of B, then, for every fixed t>0, β(At) is identical in law with sinhBt. In this paper, we extend Bougerol’s identity to an identity in law for processes up to time t, which exhibits a certain invariance of the law of Brownian motion. The extension is described in terms of anticipative transforms of B involving At as an anticipating factor. A Girsanov-type formula for those transforms is shown. An extension of a variant of Bougerol’s identity is also presented.

Keywords: Brownian motion; Exponential functional; Bougerol’s identity; Anticipative path transformation (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414922000059
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:146:y:2022:i:c:p:311-334

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.2022.01.005

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 ().