Special semimartingale

A special semimartingale is a semimartingale admitting a decompositionwith $M$ a zero-starting local martingale and $A$ a zero-starting predictable finite-variation process. Predictability of the finite-variation term is the additional requirement. With these starting conventions, this canonical decomposition is unique.

For a rate-$\lambda$ Poisson process $N$, the decomposition is $N=(N-\lambda t)+\lambda t$. The compensated part is a martingale, and the drift is deterministic, hence predictable. Although $N$ itself has finite-variation paths, assigning all of it to the finite-variation term would not give the canonical predictable decomposition.

Uniqueness follows because the difference between two such decompositions is both a local martingale and a predictable finite-variation process starting at zero, which must vanish. Predictability cannot be omitted from that argument: $N-\lambda t$ is a nonconstant local martingale with finite-variation paths on compact intervals.