Pure-jump semimartingales

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

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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 (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:	https://projecteuclid.org/euclid.bj
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:	14 Jan 2021 12:24
URI:	http://eprints.lse.ac.uk/id/eprint/108408

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