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

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### 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 $$\mathrm{L}^p$$ 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 $$\int_{0}^{t}H_s d(X_s Y_s)$$ associated with certain continuous-time martingales $$(X_t)_{t\geq 0}$$ and $$(Y_t)_{t\geq 0}$$. The process $$(X_t Y_t)_{t\geq 0}$$ is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration.

### Most cited references5

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### Maximal and singular integral operators via Fourier transform estimates

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### Problems in harmonic analysis related to curvature

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### Strong variational and jump inequalities in harmonic analysis

(2008)
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### Author and article information

###### Journal
2013-07-24
2015-02-04
###### Article
10.1016/j.jmaa.2015.02.015
1307.6403
bdc1606e-aecd-4836-a8d2-f05ac4822496

Primary 60G42, Secondary 42B15, 60G46, 60H05
J. Math. Anal. Appl. 426 (2015), 1143-1163
18 pages; v3: accepted for publication, references updated, research support acknowledged
math.PR

Probability
Probability