Generalization Of Burkholder-Davis-Gundy Inequalities

Inequalities of martingales play an important role in modern martingale theory. For example, classic Doob's inequality, good A inequalities and Burkholder-Davis-Gundy inequalities and so on, are very important results in martingale theory and the basis of discussion of the general martingale spaces. Classical Doob's inequality was extended in [1] and [2]. But many results for Burkholder-Davis-Gundy inequalities are derived only about the increasing process in Doob-Meyer decomposition of X2(X is a local martingale) and slowly increasing convex function Φ(sup x>0 xφ(x)/Φ(x) = p ≥1, is the derivative on the right of Φ). In this paper, we at first improve a series ofinequalities which can be applied to submartingales, then, by Doob-Meyer decomposition, derive some results about |X|p(p≥1, X is a martingale) and more general submartingales, and some inequalities about functions (p < 1) and logarithm functions, at last, extend these results to local martingales. Thus we generalize Burkholder-Davis-Gundy inequalities.