Pure-jump semimartingales

A new integral with respect to an integer-valued random measure is introduced. In contrast the finite variation ubiquitous in semimartingale theory, closed under stochastic integration, composition, and smooth transformations. The gives rise a previously unstudied class of pure-jump processes – sigma-locally processes. As application, it shown that every X has unique decomposition X=X0+Xqc+Xdp, where Xqc quasi-left-continuous Xdp process jumps only at predictable times, both starting zero. mirrors classical result for local martingales rigorous meaning notions continuous-time discrete-time components semimartingale. Against this backdrop, paper investigates wider are equal sum their topology constructs taxonomic hierarchy semimartingales.