Semimartingale jumps question

Question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is that it seems the author assumes these are always well defined. Is it possible that they are due to some property about jumps of semimartingales? Thanks.

Answer 1

Shiryaev and Jacod Limit theorems for stochastic processes. But I think I understand it now. Note FV means FV on each interval and we only care to define the LS integral up to each time t. If $Y$ is FV then its point masses are summable up to $t$.

So $\int_0^t |\Delta X|dY = \sum_{s\leq t} |(\Delta X)_s|dY(\{s\}) \leq \sum_1^n j_i dY(\{s\}) + \sum dY(\{s\}) < \infty$ since there are only finitely many jumps up to time $t$ of size greater than $1$.