Maximal inequalities for a series of continuous local martingales

Abstract

Let {Xj=(Xtj,ℱt),j≥1} be a sequence of continuous local martingales and {〈Xj〉} the corresponding sequence of their quadratic variation processes and let Hn(x,y),n=1,2,… be the Hermite polynomials with parametric variable y. In this paper, we consider the series ∑j=1∞Hn2(Xj,〈Xj〉) of the continuous local martingales Hn(Xj,〈Xj〉)=(Hn(Xtj,〈Xj〉t),ℱt)t≥0, j=1,2,…, and its discrete analogue, and obtain some maximal inequalities.