jumps of semimartingale form an absolute convergent series

Question

I know that every càdlàg process has only a countable number of jumps on any finite interval. Furthermore I know, that there are càdlàg processes whose jumps don't form an absolute convergent series (Are all cadlag processes of finite variation?). But when you have a look on the general Itô-Formula for a semimartingale $X$ and $f\in\mathcal{C}^2$ $$f(X_t) = f(X_0) + \int_{0+}^t f'(X_{s-})\,\mathbb{d}X_s + \frac{1}{2} \int_{0+}^t f''(X_{s-})\,\mathbb{d}[X,X]_s^c + \sum_{0<s\leq t } ( f(X_s)-f(X_{s-}) - f'(X_{s-} ) \Delta X_s)$$ you have this last sum/series, which is, as already mentioned well definied pathwise, since countable. My question is all about this sum. Is this sum absolute convergent? The equivalent question would be, is the process $$Q_t:=\sum_{0<s\leq t } ( f(X_s)-f(X_{s-}) - f'(X_{s-} ) \Delta X_s)$$ a FV-process? Can I split this series to $\sum_{0<s\leq t } ( f(X_s)-f(X_{s-}))$ and $\sum_{0<s\leq t } (- f'(X_{s-} ) \Delta X_s)$ and it is still well defined and maybe absolute convergent without any further requirements? In the book of Protter (Stochastic Integration and Defferential Equations, 2003) there is a statement (page 81) in the last sentence of the proof of the Itô-Formula (Theorem 32, page 78), which might does mean this. But I don't understand exactly what's happening there. This brought me to the question in which sense the sum/series is defined.

I know that not every semimartingale is a FV-process and for FV-processes this question is trivial. But perhaps their jumps are? Is the summation of the jumps of a semimartingale a FV-process?

Answer 1

I think I found something in George Lowther's blog, which is a kind of 'small' answer for the question I asked.

For a cadlag adapted procsess $Y$ it is equivalent that $Y$ is predictable and $\Delta Y$ is predictable. Assume $X$ is predictable. Then the processes $f(X)$, $f'(X_-)\cdot\Delta X$ and $\Delta f(X)$ are predictable as well. Therefore $Q$ is a predictable process, since $$Q_t=\sum_{0<s\leq t }\Delta Q_s.$$ According to theorem 11, the semimartingale $f(X)$ uniquely decompses into a continous local martingale starting at $0$ and in purely discontinuous semimartingale. Since the other summands are semimatringales, $Q$ is one as well. As it's countious part is constant $0$, $Q$ is a purely discontinuous semimartingale. By lemma 8 we have $Q$ is a FV-process.

Alltogether, under the additional assumption that $X$ is predictable, we now have the answer 'Yes!'.

To the last question "Is the summation of the jumps of a semimartingale $X$ a FV-process?": Again we can split the semimartingale $X$ into a continous local martingale and a purely discontinous semimartingale, which we denote by $X^d$. Then the jumps of $X$ and $X^d$ coincide. By lemma 6 we get that the summation of the jumps are a FV-process if and only if $X^d$ is a FV-process. As $X^d$ itself decompses into a FV-process $A$ and a purely discontinous local martingale $M$, it is eqivalent to $M$ beeing a FV-process. A sufficent property for this is for example, that $X$ is predictable. Because in this case $M$ would be consant $0$.

Alltogether, the jumps of semimartingale form an absolute convergent series if and only if its purely discontinous local martingale-part is an FV-process.

EDIT: For the first question I found a better answer: From Dellacherie&Meyer: PaP B (VIII.25, resp. VIII.26b) we get $Q$ is of finite Variation, without additional assumptions. Furthermore I found this post, in which Q2 is a related question.