I am looking for a good and not too difficult book that treats the following result. If $X$ is just a martingale it would be even better since I never learned semimartingales. I found it on a blog and there is a proof but I would like a book that treats the result.
Lemma : Let $X$ be a semimartingale and ${\xi}$ be a predictable, $X$-integrable process. Suppose that ${\{\tau_t\}_{t\ge 0}}$ are finite stopping times such that ${t\mapsto\tau_t}$ is continuous and increasing. Define the time-changes ${\mathcal{\tilde F}_t=\mathcal{F}_{\tau_t}}$, ${\tilde X_t=X_{\tau_t}}$ and ${\tilde \xi_t=\xi_{\tau_t}}$.
With respect to the filtration ${\mathcal{\tilde F}_t}$, ${\tilde X}$ is a semimartingale, ${\tilde\xi}$ is predictable and ${\tilde X}$-integrable, and $\displaystyle \int_0^t\tilde\xi\,d\tilde X=\int_{\tau_0}^{\tau_t}\xi\,dX$.