 Semimartingale

A semimartingale is a càdlàg adapted process expressible as $X=X_0+M+A$, where $M$ is a zero-starting local martingale and $A$ is a zero-starting adapted finite-variation process on compact time intervals. This is the class of integrators for the usual stochastic integral, subject to the integrand's integrability conditions.

Every Itô process is a continuous semimartingale. A Poisson process is also a semimartingale, with jumps and paths of finite variation. For a unit-rate Poisson process $N$, the decomposition $N=(N-t)+t$ separates a compensated Poisson process from its deterministic rate. A general adapted semimartingale decomposition need not be unique; requiring a predictable finite-variation part gives uniqueness for a special semimartingale.