Maximal Inequalities for Martingales and Their Differential Subordinates

̧kowski, Adam Ose

doi:10.1007/s10959-012-0458-8

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Cited by 4 publications

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“…Assuming equality, we obtain the function which coincides with the special function of Section 2 on the sets with the function F and the parameters κ 1 , κ 2 to be found. Expressions of this type appear in many Burkholder's functions (see [11], [12] and [13]); actually, the formulas on D 1 and D 2 are also of similar type. The unknown parameters can be derived from the fact that U is of class C 1 ; the luck is with us, we are led precisely to the right formula.…”

Section: Derivingmentioning

confidence: 99%

A sharp maximal inequality for continuous martingales and their differential subordinates

Oşekowski

2013

Czech Math J

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Section: Derivingmentioning

confidence: 99%

A sharp maximal inequality for continuous martingales and their differential subordinates

Oşekowski

2013

Czech Math J

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Long-Range Dependence as a Phase Transition

Samorodnitsky

2016

Springer Series in Operations Research and Financial Engineering

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A weighted maximal inequality for differentially subordinate martingales

Bañuelos

Oşekowski

2018

Proc. Amer. Math. Soc.

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The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that f = ( f n ) n ≥ 0 f=(f_n)_{n\geq 0} , g = ( g n ) n ≥ 0 g=(g_n)_{n\geq 0} are martingales such that g g is differentially subordinate to f f , and let w = ( w n ) n ≥ 0 w=(w_n)_{n\geq 0} be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by M f = sup n ≥ 0 | f n | Mf=\sup _{n\geq 0}|f_n| , M w = sup n ≥ 0 w n Mw=\sup _{n\geq 0}w_n the maximal functions of f f and w w , we prove the weighted inequality | | g | | L 1 ( w ) ≤ C | | M f | | L 1 ( M w ) , \begin{equation*} ||g||_{L^1(w)}\leq C||Mf||_{L^1(Mw)}, \end{equation*} where C = 3 + 2 + 4 ln ⁡ 2 = 7.186802 … C=3+\sqrt {2}+4\ln 2=7.186802\ldots . The proof rests on the existence of a special function enjoying appropriate majorization and concavity.

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Maximal Inequalities for Martingales and Their Differential Subordinates

Abstract: We introduce a method of proving maximal inequalities for Hilbertspace-valued differentially subordinate local martingales. As an application, we provewhere β = 2.585 . . . is the best possible.

Cited by 4 publications

References 19 publications

A sharp maximal inequality for continuous martingales and their differential subordinates

A sharp maximal inequality for continuous martingales and their differential subordinates

Long-Range Dependence as a Phase Transition

A weighted maximal inequality for differentially subordinate martingales

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