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 $\mathbb{E}{\left[M\right]}_{\mathit{\tau }}^{q∕2}$, for any stopping time τ 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.