  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.
