Open Access
2021 On absolute continuity and singularity of multidimensional diffusions
David Criens
Electron. J. Probab. 26: 1-26 (2021). DOI: 10.1214/20-EJP555

Abstract

Consider two laws P and Q of multidimensional possibly explosive diffusions with common diffusion coefficient a and drift coefficients b and b+ac, respectively, and the law P of an auxiliary diffusion with diffusion coefficient c,ac1a and drift coefficient c,ac1b. We show that PQ if and only if the auxiliary diffusion P explodes almost surely and that PQ if and only if the auxiliary diffusion P almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.

Citation

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David Criens. "On absolute continuity and singularity of multidimensional diffusions." Electron. J. Probab. 26 1 - 26, 2021. https://doi.org/10.1214/20-EJP555

Information

Received: 30 April 2020; Accepted: 12 November 2020; Published: 2021
First available in Project Euclid: 19 January 2021

Digital Object Identifier: 10.1214/20-EJP555

Subjects:
Primary: 60G44 , 60H10 , 60J60 , 91B70

Keywords: Absolute continuity , explosion , Integral test , multidimensional diffusion , perpetual integral , Random time change , singularity , uniformly integrable martingale

Vol.26 • 2021
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