Pure-jump semimartingales

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

$$\mathit{X}={\mathit{X}}_0+{\mathit{X}}^{\mathrm{qc}}+{\mathit{X}}^{\mathrm{dp}},$$

where ${\mathit{X}}^{\mathrm{qc}}$ is quasi-left-continuous and ${\mathit{X}}^{\mathrm{dp}}$ 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 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.