# The Bellman function technique for multilinear martingale estimates

Škreb, Kristina Ana (2017) The Bellman function technique for multilinear martingale estimates. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

This thesis develops the Bellman function technique needed in the proofs of the $$L^p$$ estimates in the context of two or three different martingales. If the martingales are adapted to the same filtration, the corresponding Bellman function, which satisfies certain convexity-type properties, is constructed on the domain determined by finitely many control parameters. The constructed function is used to prove $$L^p$$ estimates for the martingale paraproduct (both in discrete-time and in continuous-time) and to give an alternative proof of $$L^p$$ boundedness of the heat flow paraproducts. On the other hand, if the martingales are adapted with respect to different filtrations, a certain Bellman-type expression (a control process) which imitates the properties of Bellman functions is constructed. Several novel martingale estimates are proved and then applied in different branches of mathematics. New $$L^p$$ estimates are established for the general twisted paraproduct with respect to two general multiplicative groups of dilations. Also, a possible direction in which Itō’s integration theory can be extended beyond the limitations of the Bichteler–Dellacherie theorem is presented. The stochastic integral is constructed in a specific context, where the integrator is not necessarily a semimartingale. The thesis is organized as follows. Chapter 1 (“Introduction”) introduces the Bellman function technique and explains the objectives and hypotheses of this research. In Chapter 2 (“Preliminaries”) we give some preliminary results and definitions regarding conditionals expectations, martingales and stochastic integration. In Chapter 3 (“Bellman function and $$L^p$$ estimates for dyadic paraproducts”) we give an explicit formula for one possible Bellman function associated with the $$L^p$$ boundedness of dyadic paraproducts regarded as trilinear forms. The constructed function has a similar form to the one constructed by F. Nazarov and S. Treil in [46] (for the problem of Haar multipliers). For a special choice of exponents, we also construct simpler Bellman functions that give us asymptotic behavior of the constants. In Chapter 4 (“$$L^p$$ estimates for paraproducts”) we apply the same Bellman function (constructed in the previous chapter) in various other settings, to give self-contained alternative proofs of the $$L^p$$ estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows. In Chapter 5 (“$$L^p$$ estimates for the generalized martingale transform”) 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. To do so we construct an appropriate control process with required “convexity” properties. In Chapter 6 (“Twisted paraproducts and stochastic integrals”) we apply the martingale inequalities obtained in Chapter 5 to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. That way new $$L^p$$ estimates are established for paraproducts with respect to two different dilation structures. As another application we construct stochastic integrals $$\int_0^t H_sd(X_sY_s)$$ associated with certain continuous-time martingales $$(X_t)_{t \geqslant 0}$$ and $$(Y_t)_{t \geqslant 0}$$. The process $$(X_tY_t)_{t \geqslant 0}$$ is shown to be a “good integrator”, although it is not necessarily a semimartingale, or even adapted to any convenient filtration. Finally, in Chapter 7 (“Noncommutative martingale paraproducts”) we present $$L^p$$ estimates for martingale paraproducts of dyadic algebra-valued martingales. For a special choice of exponents we construct the Bellman function that yields estimates with sharp constants.