Applications of pathwise Burkholder–Davis–Gundy inequalities

Abstract

In this paper, after generalizing the pathwise Burkholder–Davis–Gundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities

1 to the Bessel process of order $\alpha\geq1$

2 to the case of a random exponent $p$

3 to martingales stopped at a time $\tau$ which belongs to a well studied class of random times