On the Iterated Martingale Transforms

Abstract

Let f = (f_n ,F_n ) _n≥0 be a martingale on some filtered complete probability space (Ω,F,P ) with the usual conditions. We define the iterated martingale transforms I ^(m)(f ) = (I_n^(m), (F_n )) (m ≥1) with respect to f, the discrete analogues of the iterated stochastic integrals. We obtain the L^p (1≤p)-estimates of I ^(m)(f )

||I ^(m)*||_p ≤ (4mp)^m ||S ^m (f )||_p

and we also characterize a continuous martingale by the limit of the iterated martingale transforms.