Ito's formula for Poisson process

Question

Suppose ($Y_t$) is a rate 1 Poisson process, and consider the jump process $Z_t=Y_{\int_0^tf(X_s)ds}$ for some non-negative process $X_s$. What would be the quadratic variation of $Z$, and how would one write Ito's formula for $Z$? Please give references to books/ theorems used if possible.

Thanks

Answer 1

For Poisson process, the quadratic variation is just $N(t)$. For $Z_t$, the quadratic variation is just $N(\int_0^tf(X_s)ds)$, if $f$ is positive. Without positive assumption, the 'time' $\int_0^tf(X_s)ds$ can be negative