Applications of pathwise Burkholder-Davis-Gundy inequalities

We present several applications of the pathwise Burkholder-Davis-Gundy (BDG) inequalities. Most importantly we prove them for cadlag semimartingales and a general function $\Phi$, and use this to derive BDG inequalities (non-pathwise ones) for the Bessel process of order $\alpha\geq 1$ and for martingales stopped at $\tau$, with $\tau$ in a well studied class of random times.