A generalisation of the Burkholder-Davis-Gundy inequalities

Abstract

{Consider a càdlàg local martingale $M$ with square brackets $[M]$. In this paper, we provide upper and lower bounds for expectations of the type $\mathbb{E}[M{]}_{\tau }^{q/2}$, 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 $|M^{\ast }{|}^q$ 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.

8 pages