I am reading a paper where they start by assuming some process follows $$ \frac{dX_t} {X_{t-}} = \alpha_t dt + \sqrt{V_t} dW_t + \int_{x > -1} x \tilde{\mu}(dt, dx) $$ with $\alpha_t$ and $V_t$ cadlag, $W$ a Brownian motion, $\mu$ an integer valued random measure with compensator $\nu(dt,dx) = a_t dt \otimes \nu(dx)$ for some process and Levy measure $\nu$ and the associated martingale measure is $\tilde{\mu} = \mu - \nu$.
To try and understand this I have (tried to) read the relevant parts in Limit theorems by Jacod and Shiryaev and Discretization of processes by Protter and Jacod, so I can see the similarities to what they call Îto semimartingales, but I just can make any sense to the lower limit. Is this just assuming the process follows an Îto semimartingale or is it something more?
Edit: Its a version of this paper: https://economics.indiana.edu/home/about-us/events/conferences-and-workshops/2013/files/2013-11-05-01.pdf see page 4.