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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.)

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    Really, nothing on this? Anyone have suggestions for a better way to pose my question or different forum? Is this more appropriate for mathoverflow?2012-04-22
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    This is fine for math.stackexchange. I'll answer this properly later this week if nobody else does. Btw, your Hypothesis A is missing some absolute-value signs. If you put them in then the hypothesis is equivalent to $X^d$ being a finite variation process. This is *not* satisfied by all semimartingales (e.g., Cauchy process). Q1 does not have a good answer though, it's just true when it is true. In general there is no reason to expect this to hold.2012-04-25
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    Good catch! I added the absolute value signs in the question. For Q1, I'd be content with an example for when it holds and when it doesn't. I'm really just trying to wrap my head around what these different processes are supposed to *look like*.2012-04-26
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    @GeorgeLowther: If you add a short answer below, I'll almost surely accept it. Still have something? :)2012-05-01

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