Pure-jump semimartingales

TL;DR

This paper introduces a new integral for pure-jump processes, enabling a unique decomposition of semimartingales into continuous and pure-jump parts, and develops a classification hierarchy for pure-jump semimartingales.

Contribution

It presents a novel integral that is closed under key operations and defines a new class of pure-jump processes, expanding the theoretical framework of semimartingale decomposition.

Findings

Established a unique semimartingale decomposition into continuous and pure-jump components.

Introduced a new class of sigma-locally finite variation pure-jump processes.

Developed a hierarchy classifying 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 (Jacod and Shiryaev, 2003, 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 $$X = X_0 + X^{\mathrm{qc}}+X^{\mathrm{dp}},$$ where $X^{\mathrm{qc}}$ is quasi-left-continuous and $X^{\mathrm{dp}}$ 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 (Yoeurp, 1976, 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.