Optional time-change theorem

Let $M$ be a continuous local martingale and let $({\tau }_s)$ be an increasing continuous family of finite stopping times. Under the usual compatibility conditions, $M_{{\tau }_s}$ is a continuous local martingale for the time-changed filtration ${\mathcal{F}}_{{\tau }_s}$. A finite-variation process remains of finite variation after the same time change.