= Semimartingale decomposition
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>.
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