 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.