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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.

A semimartingale admits $X=X_0+M+A$ with $M_0=A_0=0$, $M$ a local martingale, and $A$ an adapted finite-variation process. These initial-value normalizations alone do not make the decomposition unique when jumps are allowed.

For a unit-rate Poisson process $N$, both $N=0+N$ and $N=(N-t)+t$ are such decompositions: $N-t$ is a local martingale that also has finite variation. If a decomposition can be chosen with $A$ predictable, $X$ is a special semimartingale, and its predictable decomposition is unique. For a continuous semimartingale, requiring both parts continuous gives the continuous semimartingale decomposition, unique because a continuous finite-variation local martingale is constant.