A generalisation of the Burkholder-Davis-Gundy inequalities

Saul Jacka, Ma. Elena Hérnandez-Hérnandez

Published 2020-09-30Version 1

Consider a c\'adl\'ag local martingale $M$ with square brackets $[M]$. In this paper, we provide lower and upper bounds for expectations of the type $E [M]^{q/2}_{\tau}$, for any stopping time $\tau$ and $q\ge 2$. This result is a Burkholder-Davis-Gundy-type inequality as it relates 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 of convex moderate functions is also treated.