Applications of pathwise Burkholder-Davis-Gundy inequalities

TL;DR

This paper extends the pathwise Burkholder-Davis-Gundy inequalities to cadlag semimartingales and general functions, and applies these results to derive classical BDG inequalities for Bessel processes and stopped martingales.

Contribution

It proves the pathwise BDG inequalities for broader classes of processes and functions, and derives new BDG inequalities for Bessel processes and martingales stopped at random times.

Findings

BDG inequalities proven for cadlag semimartingales and general functions

BDG inequalities derived for Bessel processes of order ≥ 1

BDG inequalities established for martingales stopped at certain random times

Abstract

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.