A sharp maximal inequality for continuous martingales and their differential subordinates

Adam Osękowski

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 1001-1018
  • ISSN: 0011-4642

Abstract

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Assume that X , Y are continuous-path martingales taking values in ℝ ν , ν ≥ 1 , such that Y is differentially subordinate to X . The paper contains the proof of the maximal inequality ∥ sup t ≥ 0 | Y t | ∥ 1 ≤ 2 ∥ sup t ≥ 0 | X t | ∥ 1 . The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

How to cite

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Osękowski, Adam. "A sharp maximal inequality for continuous martingales and their differential subordinates." Czechoslovak Mathematical Journal 63.4 (2013): 1001-1018. <http://eudml.org/doc/260767>.

@article{Osękowski2013,
abstract = {Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb \{R\}^\nu $, $\nu \ge 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality \[ \Vert \sup \_\{t\ge 0\} |Y\_t| \Vert \_1\le 2\Vert \sup \_\{t\ge 0\} |X\_t| \Vert \_1. \] The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.},
author = {Osękowski, Adam},
journal = {Czechoslovak Mathematical Journal},
keywords = {martingale; stochastic integral; maximal inequality; differential subordination; martingale; stochastic integral; maximal inequality; differential subordination},
language = {eng},
number = {4},
pages = {1001-1018},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A sharp maximal inequality for continuous martingales and their differential subordinates},
url = {http://eudml.org/doc/260767},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Osękowski, Adam
TI - A sharp maximal inequality for continuous martingales and their differential subordinates
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1001
EP - 1018
AB - Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb {R}^\nu $, $\nu \ge 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality \[ \Vert \sup _{t\ge 0} |Y_t| \Vert _1\le 2\Vert \sup _{t\ge 0} |X_t| \Vert _1. \] The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.
LA - eng
KW - martingale; stochastic integral; maximal inequality; differential subordination; martingale; stochastic integral; maximal inequality; differential subordination
UR - http://eudml.org/doc/260767
ER -

References

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