The Annals of Probability

Differential Subordination and Strong Differential Subordination for Continuous-Time Martingales and Related Sharp Inequalities

Gang Wang

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Abstract

Let $X$ and $Y$ be two continuous-time martingales. If quadratic variation of $X$ minus that of $Y$ is a nondecreasing and nonnegative function of time, we say that $Y$ is differentially subordinate to $X$ and prove that $\|Y\|_p \leq (p^\ast - 1)\|X\|_p$ for $1 < p < \infty$, where $p^\ast = p \vee q$ and $q$ is the conjugate of $p$. This inequality contains Burkholder's $L^p$-inequality for stochastic integrals, which implies that the above inequality is sharp. We also extend his concept of strong differential subordination and several other of his inequalities, and sharpen an inequality of Banuelos.

Article information

Source
Ann. Probab., Volume 23, Number 2 (1995), 522-551.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988278

Digital Object Identifier
doi:10.1214/aop/1176988278

Mathematical Reviews number (MathSciNet)
MR1334160

Zentralblatt MATH identifier
0832.60055

JSTOR
links.jstor.org

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60G42: Martingales with discrete parameter 60G46: Martingales and classical analysis

Keywords
Differential subordination martingale sharp inequalities strong differential subordination submartingale

Citation

Wang, Gang. Differential Subordination and Strong Differential Subordination for Continuous-Time Martingales and Related Sharp Inequalities. Ann. Probab. 23 (1995), no. 2, 522--551. doi:10.1214/aop/1176988278. https://projecteuclid.org/euclid.aop/1176988278


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