Suppose that $Y(t)$ is a Lévy process, $\tau(t)=\int_0^t e^{-\alpha Y(s)}ds,$ and $Z(t)$ is an $(\alpha + 1)$-stable Levy process, where $0<\alpha<1$. Then according to the self-similar property, $$X(t)=\int_0^t e^{\frac{\alpha}{\alpha + 1}Y(s)}dZ(\tau(s))$$ is also an $(\alpha+1)$-stable Lévy process.
May anyone show me the details please? I am confused with the time transformation. Also I wonder how to complement Ito's formula with a time transformation, for example, to calculate $dZ(\tau(s))=d\int_0^{\tau(t)}\int z\tilde{N}(ds,dz)$, where $\tilde{N}$ is a compensated Poisson random measure. Is there a theorem parallel to the Newton-Leibniz formula in stochastic calculus? Many thanks!!!