Consider a càdlàg local martingale M with square brackets . In this paper, we provide upper and lower bounds for expectations of the type , for any stopping time τ and , 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 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.
Saul Jacka gratefully acknowledges funding received from the Alan Turing Institute for their financial support under the EPSRC grant EP/N510129/1.
"A generalisation of the Burkholder-Davis-Gundy inequalities." Electron. Commun. Probab. 27 1 - 8, 2022. https://doi.org/10.1214/22-ECP493