A generalisation of the Burkholder-Davis-Gundy inequalities

TL;DR

This paper extends Burkholder-Davis-Gundy inequalities to provide bounds for expectations of local martingales' square brackets, relating maximums to predictable processes, applicable for various stopping times and jump behaviors.

Contribution

It introduces a generalized inequality framework for local martingales, linking expectations of their quadratic variation to predictable projections, broadening the classical BDG inequalities.

Findings

Established bounds for expectations of quadratic variation in terms of predictable processes.

Extended the inequalities to include moderate functions.

Applicable to a wide class of local martingales with jumps.

Abstract

{Consider a c\`adl\`ag local martingale $M$ with square brackets $[M]$. In this paper, we provide upper and lower bounds for expectations of the type ${\mathbb E} [M]^{q/2}_{\tau}$, for any stopping time $\tau$ and $q\ge 2$, in terms of predictable processes. This result can be thought of as a Burkholder-Davis-Gundy type inequality in the sense that it can be used to relate the expectation of the running maximum $|M^*|^q$ to the expectation of the dual previsible projections of the relevant powers of the associated jumps of $M$. The case for a class of moderate functions is also discussed.