Illinois Journal of Mathematics

Sharp maximal $L^{p}$-estimates for martingales

Rodrigo Bañuelos and Adam Osȩkowski

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Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_{p}$, $C_{p}$ and $\mathfrak{c}_{p}$ such that

\begin{eqnarray*}\sup_{t\geq0}\big\|X_{t}\big\|_{p}&\leq&C_{p}\big\|-\inf_{t\geq0}X_{t}\big\|_{p},\\\phantom{\Vert }\Vert \sup_{t\geq0}X_{t}\Vert _{p}&\leq&c_{p}\|-\inf_{t\geq0}X_{t}\|_{p}\end{eqnarray*} and

\[\Vert \sup_{t\geq0}\vert X_{t}\vert \Vert _{p}\leq\mathfrak{c}_{p}\|-\inf_{t\geq0}X_{t}\|_{p}.\] The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities are deduced from the existence of special functions, enjoying certain majorization and convexity-type properties. Some applications concerning harmonic functions on Euclidean domains are indicated.

Article information

Illinois J. Math., Volume 58, Number 1 (2014), 149-165.

First available in Project Euclid: 1 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G44: Martingales with continuous parameter 31B05: Harmonic, subharmonic, superharmonic functions


Bañuelos, Rodrigo; Osȩkowski, Adam. Sharp maximal $L^{p}$-estimates for martingales. Illinois J. Math. 58 (2014), no. 1, 149--165. doi:10.1215/ijm/1427897172.

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