Initial jump of semimartingale integrator I've read that it is a common assumption in the literature on stochastic integration that a semimartingale integrand $S$ may jump at $t = 0$ and a common convention is to assume $S_{0-} = 0$, so the stochastic integral doesn't depend on on the initial jump of the integrator. Why is this important/convenient? If there is P-a.s. no initial jump of the integrator is this still useful?
 A: I'm not absolutely sure about it, but this might be an answer:
It relevant in the semimartingale context of integration, because the integral considers also the initial values of the involved processes. For $X$ semimartingale and $H$ adapted caglad it holds
$$H\bullet X_t:=\int_0^tH_s\, \mathbb{d}X_s=H_0X_0+\int_{0+}^tH_s\, \mathbb{d}X_s.$$
If you now consider the jumps $\Delta Y := Y-Y_-$ of a process $Y$ at the time $t=0$ you have to declare what $Y_{0-}$ is, since it isn't defiened yet. You probably now the theorem that the processes $\Delta (H\bullet X)_t$ and $H_t\cdot \Delta X_t$ are indistinguishable. With the convention $Y_{0-}=0$ for every Semimartingale, everything is fine. Othewise you have to define $$H\bullet X_{0-}:=H_0\cdot (X_{0-}-X_0)+H\bullet X_0=H_0X_{0-}.$$
Furthermore, you want to have a 'nice' formula of partial integration. With the convention you can just write/define $$[Y,X]:=XY-\int X_-\,\mathbb{d}Y-\int Y_-\,\mathbb{d}X$$ and meaning $$[Y,X]_t=X_tY_t-\int_0^t X_-\,\mathbb{d}Y-\int_0^t Y_-\,\mathbb{d}X$$ or $$[Y,X]_t=X_tY_t-\int_{0+}^t X_-\,\mathbb{d}Y-\int_{0+}^t Y_-\,\mathbb{d}X$$ and it doesn't matter, since it's both the same. Without the convention you have to define it by the latter one, since you want to hold $[X,Y]_0=X_0Y_0$ and not $[X,Y]_0=X_0Y_0-X_{0-}Y_0-Y_{0-}X_0$. And without the convention you would have to definie explicit again what you mean by $[X,Y]_{0-}$. For reasons you want to hold $\Delta [X,Y]=\Delta X \Delta Y$, since it holds for every time $t>0$ and you dont want to 'destroy' this nice identity for the time $t=0$, it would be $$[X,Y]_{0-}:=X_0Y_0-\Delta X_0 \Delta Y_0=X_{0-}Y_{0-}+X_{0-}\Delta Y_0+Y_{0-}\Delta X_0.$$
As you see, with the convention both sides would be $0$ and you don't have to care for anything.
