1. A sharp maximal inequality for continuous martingales and their differential subordinates
- Creator:
- Osekowski, Adam
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- martingale, stochastic integral, maximal inequality, and differential subordination
- Language:
- English
- Description:
- Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu $, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$ \|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1. $$ 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public