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November 2021 Pure-jump semimartingales
Aleš Černý, Johannes Ruf
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Bernoulli 27(4): 2624-2648 (November 2021). DOI: 10.3150/21-BEJ1325

Abstract

A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory, the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes – the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale X has a unique decomposition

X=X0+Xqc+Xdp,

where Xqc is quasi-left-continuous and Xdp is a sigma-locally finite variation pure-jump process that jumps only at predictable times, both starting at zero. The decomposition mirrors the classical result for local martingales and gives a rigorous meaning to the notions of continuous-time and discrete-time components of a semimartingale. Against this backdrop, the paper investigates a wider class of processes that are equal to the sum of their jumps in the semimartingale topology and constructs a taxonomic hierarchy of pure-jump semimartingales.

Acknowledgements

The authors would like to thank an anonymous referee for critical comments that have lead to several improvements in the paper.

Citation

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Aleš Černý. Johannes Ruf. "Pure-jump semimartingales." Bernoulli 27 (4) 2624 - 2648, November 2021. https://doi.org/10.3150/21-BEJ1325

Information

Received: 1 April 2020; Revised: 1 October 2020; Published: November 2021
First available in Project Euclid: 24 August 2021

Digital Object Identifier: 10.3150/21-BEJ1325

Rights: Copyright © 2021 ISI/BS

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Vol.27 • No. 4 • November 2021
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