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2018 Absolute continuity of semimartingales
David Criens, Kathrin Glau
Electron. J. Probab. 23(none): 1-28 (2018). DOI: 10.1214/18-EJP238

Abstract

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov’s theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Itô-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multidimensional Itô-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

Citation

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David Criens. Kathrin Glau. "Absolute continuity of semimartingales." Electron. J. Probab. 23 1 - 28, 2018. https://doi.org/10.1214/18-EJP238

Information

Received: 15 October 2018; Accepted: 23 October 2018; Published: 2018
First available in Project Euclid: 19 December 2018

zbMATH: 07021681
MathSciNet: MR3896862
Digital Object Identifier: 10.1214/18-EJP238

Subjects:
Primary: 60G44, 60G48

JOURNAL ARTICLE
28 PAGES


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Vol.23 • 2018
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