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I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have

$dS_t=S_{t-}\mu+S_{t-}\sigma dB_t +S_{t-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq t\leq T$

where $N$ is a counting measure whose mean measure is $\nu(dy)dt$.

Define $q_t=\frac{1}{rT}(1-e^{-r(T-t)})$, and $X_0=q_0S_0-e^{-rT}K$ where $0<r, K$ are given real numbers, let $X_t$ be a process that satisfies

$ dX_t=q_tdS_t+r(X_{t-}-q_tS_{t-})dt\;\;0\leq t\leq T$. Define $Z_t=\frac{X_t}{S_t}$

1) I would like to know what is $dZ_t$ ??

2) What is the intuitive meaning of the mean measure $\nu(dy)dt$??

I know that I need to use Ito's product rule but I am still confused about jump processes. Could someone help me with this problem? Thank you so much.

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