Semimartingale

A semimartingale is the sum of a local martingale and an adapted finite-variation process. This is the broad class of integrators for which the Itô stochastic integral is defined.

A process $A$ has finite variation on compact intervals when every sample path has finite total variation of a function there. Such a process can be integrated pathwise by the Lebesgue-Stieltjes integral.

A semimartingale has a decomposition $X=X_0+M+A$ into a local martingale $M$ and an adapted finite-variation process $A$. Under standard normalizations the decomposition is unique.

A continuous semimartingale has a decomposition in which both the local-martingale and finite-variation parts are continuous.