Definition of semimartingale and stochastic integration Hey I found two definitions of semimartingale. One states that semimartingale is a sum of local martingale and FV process and the second one states that semimartingale is a sum of FV process and locally square integrable martingale. Are these definitions equivalent? I would like to define stochastic integral of caglad process with respect to semimartingale. Can I define this integral as the sum of Lebesgue-Stieltjes integral and integral w.r.t locally square integrable process? Or I have to define this integral using the first definition? What is the difference? Is this integral well defined? I would be thankful for any help.
 A: Regarding your two definitions: I know the first one for classical semimartingales (or just semimartingales). The second one is the defintion of decomposable processes (e.g. in the book of Protter Ch.II,3).
The two definitions are equivalent: As every process which is locally a square integrable martingale is of course also a local martingale, all processes satisfying the second definition also satisfy the first definition. For the reverse you have to use the Fundamental Theorem of Local Martingales (e.g. in the book of Protter Ch.III,6,Th.25). This states that every local martingale can be written as the sum of two local martingales, whereas one of them is a FV process and the other has bounded jumps. The local martingale with bounded jumps is a locally bounded martingale und thus locally a square integrable martingale.
To your second question: As the FV part can be used for a Lebesgue-Stieltjes integral the answer is YES (for a FV process the stochastic and the Lebesgue-Stieltjes integral are indistinguishable, see Protter Ch.II,5,Th.17). Since you are only integrating càglàd processes the integral is well defined.
