Question
We have a stochastic process $(X_t)_t$ that can be written by: $$ dX_t = -\mu dt - \sigma dB_t + d\nu_t, $$ with $(B_t)_t$ a standard brownian motion and $d\nu_t$ equal to some number $\xi_t$ at some finite number of times $\eta_t$ and $0$ otherwise. Let $V$ be an arbitrary function then we want to apply Ito's formula to the expression: $$ V(x) - \mathbb{E}\left[ V(X_{\Delta t}) e^{-\alpha\Delta t } \right] $$ to obtain that this expression is equal to $$ \Delta t \left( -\frac{1}{2} \sigma^2 \frac{d^2V}{dx^2} + \mu \frac{dV}{dx} + \alpha V \right) $$ or at least that's what I think has to be shown.
Origin Of The Question
At the top of page 309 of Inventory Management With Stochastic Lead Times Ito's formula is used but I don't understand this step.