## The Annals of Probability

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

Gang Wang

#### 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

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

#### 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