We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob–Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder–Davis–Gundy inequalities. A connection with some balayage formulae is also established.
"Burkholder’s submartingales from a stochastic calculus perspective." Illinois J. Math. 52 (3) 815 - 824, Fall 2008. https://doi.org/10.1215/ijm/1254403716