A generalisation of the Burkholder-Davis-Gundy inequalities

Consider a cadlag local martingale M with square brackets [M]. In this paper, we provide upper and lower bounds for expectations of the type E [M](q/2)(tau), for any stopping time tau and q >= 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 vertical bar M*vertical bar(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.