Doob-Type Estimates for Differentially Subordinated Martingales

Abstract

Let X, Y be local martingales such that Y is differentially subordinate to X. The article contains the proof of the sharp estimates

As an application, we establish sharp inequalities for stochastic integrals with respect to Brownian motion in ℝ ^d , derive some maximal bounds for harmonic functions and Riesz system and present tight estimates for a stopped three-dimensional Bessel process.

Keywords:

• Bessel process
• Differential subordination
• Harmonic function
• Martingale
• Maximal inequality
• Riesz system
• Stochastic integral

Mathematics Subject Classification:

• Primary 60G44
• Secondary 31B05

Acknowledgments

This work was partially supported by the Foundation for Polish Science and MNiSW Grant N N201 397437.