One modification of the martingale transform and its applications to paraproducts and stochastic integrals

Abstract

In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected L p estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals t 0 Hsd(XsYs) associated with certain continuous-time martingales (Xt) t≥0 and (Yt) t≥0 . The process (XtYt) t≥0 is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration.2010 Mathematics Subject Classification. Primary 60G42; Secondary 42B15, 60G46, 60H05.

Norm variation of ergodic averages with respect to two commuting transformations

“…The range of exponents p 1 , p 2 , p, ̺ in the above discussion is likely not exhausted as the analogous work [24] in the simplified setting suggests. This paper, while self-contained, builds on a technique for bounding multi-linear and multiscale singular integral operators gradually developed by the authors in [14], [15], [22], [23], [24], [25], [26]. We consider the present application to quantitative norm convergence for double ergodic averages a milestone in these efforts.…”

Density theorems for anisotropic point configurations

“…It could be interesting to study (6.3) for more general dilation structures, i.e., when K is a generalized Calderón-Zygmund kernel, such as in the case (6.2), which is relevant here. In this context, a similar but still different object has been studied by Škreb and the present author [42]. Some generalizations of the result by Durcik [12] are straightforward: the single L 2 n × • • • × L 2 n bound for (6.3) can be extracted easily from the presented proof of Theorem 2 combined with a cone decomposition of the kernel.…”

Convergence of ergodic-martingale paraproducts

“…The expression in (2.7) could be called the martingale-martingale paraproduct. We emphasize that it is different from the more classical martingale paraproducts appearing in [1,4,6,24,26], because conditional expectations with respect to different filtrations are applied to the functions F and G. However, Škreb and one of the present authors [25] have already studied this object to some extent, which is what we find convenient below.…”

“…The same arguments also apply here after (U i ) ∞ i=0 and (V i ) ∞ i=0 have been reduced as in the previous paragraph; one only needs to redefine the Haar functions to accommodate for possibly uneven splitting of atoms. This particular case of (2.7) is the most difficult ingredient in our proof; we encourage the reader to go over the details in either [25] or [23].…”

“…In probability theory, paraproducts of two martingales appear as variants of Burkholder's martingale transforms [4]. Development of their theory, parallel to the one of analytical paraproducts, was initiated by Bañuelos and Bennett [1] (in continuous time) and Chao and Long [6] (in discrete time); also see more recent papers [24,25,26]. Here we take even more liberty with usage of the word, as we only have two alike objects that add up to the pointwise product (1.4), modulo the trivial term f g. Now we formulate the main result of this note.…”