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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 $\mathfrak {a}$ and drift coefficients $\mathfrak {b}$ and $\mathfrak {b}+ \mathfrak {a}\mathfrak {c}$, respectively, and the law $P^{\circ }$ of an auxiliary diffusion with diffusion coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {a}$ and drift coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {b}$. We show that $P \ll Q$ if and only if the auxiliary diffusion $P^{\circ }$ explodes almost surely and that $P\perp Q$ if and only if the auxiliary diffusion $P^{\circ }$ 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

JOURNAL ARTICLE
26 PAGES


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