From Kal97, pg. 446:
Theorem 23.14 Any semimartingale $X$ has an a.s. unique decomposition $X=X_0 + X^c + X^d$ where $X^c$ is a continuous local martingale with $X_0^c=0$ and $X^d$ is a purely discontinuous semimartingale.
Q1: When is it true that
$X_t^d = \sum_{0 \leq u \leq t} \Delta X_u$
(where $\Delta X_t \equiv X_t - X_{t-}$ and $X_{t-} \equiv \lim_{u \nearrow t} X_u$)?
Is it always true?
Pro05, pg.221 seems to require:
Hypothesis A. $\sum_{0 \leq u \leq t} | \Delta X_u | \lt \infty$ a.s., each $t \gt 0$
But I thought this was trivially satisfied by semimartingales since their quadratic variation exists and is finite.
Q2: When is it true that $X^d$ is a finite variation process?
I suppose if someone answers my first question the second is necessarily answered?
(Thx for bearing with me. This material is way over my head. Just formulating a good question has been a challenge.)