Time change of a continuous process

If $X$ has continuous paths and ${\tau }_s$ is a finite continuous increasing family of random times, then $X_{{\tau }_s}$ also has continuous paths. A strictly increasing continuous clock that tends to infinity has a continuous inverse.

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.