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 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 (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.

Publication Type:	Article
Additional Information:	© International Statistical Institute/Bernoulli Society for Mathematical Statistics and Probability
Publisher Keywords:	jump measure, Lévy process, predictable compensator, semimartingale topology, stochastic calculus
Subjects:	H Social Sciences > HG Finance Q Science > QA Mathematics
Departments:	Bayes Business School > Finance

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