Continuous semimartingale
= Continuous semimartingale
A continuous <semimartingale> is a continuous adapted process admitting a decomposition $X=X_0+N+V$, with $N$ a zero-starting <continuous local martingale> and $V$ continuous adapted <finite variation>. The decomposition is unique because a continuous finite-variation local <martingale> starting at zero vanishes. Its <quadratic variation> is $[X]=[N]$.