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Pure-jump semimartingales

Černý, Aleš and Ruf, Johannes (2021) Pure-jump semimartingales. Bernoulli. ISSN 1350-7265 (In Press)

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A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev [6, II.1.5]), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of purejump 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 + X qc + X dp , where X qc is quasi-left-continuous and X dp 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 (Yoeurp [12, Theoreme 1.4]) 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.

Item Type: Article
Official URL:
Additional Information: © 2021 International Statistical Institute/Bernoulli Society for Mathematical Statistics and Probability
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Jan 2021 16:24
Last Modified: 17 Mar 2021 00:23

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