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Title:Bounds on the size of strong subordinates of submartingales and subharmonic functions
Author(s):Hammack, William
Doctoral Committee Chair(s):Peck, N.T.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Suppose X is a submartingale that is continuous on the right with limits from the left and H is a predictable process bounded by 1 in absolute value. Let $Y = (Y\sb{t})\sb{t\ge 0}$ where$$Y\sb{t} = H\sb0X\sb0 + \int\sb{(0,t\rbrack} H\sb{s}dX\sb{s}.$$An interesting and important question is: How large is Y compared to X? While it is impossible to give general $L\sp{p}$-inequalities for $p > 1,$ we show that there are sharp weak-type inequalities, and under the additional assumption that X is bounded, sharp bounds on the distribution of the maximal function $Y\sp\*$ of $Y.$ For example, for all $\lambda > 0$,$$\lambda P(Y\sp\*\ge\lambda)\le 6\Vert X\Vert\sb1$$and the constant 6 is the best possible. In fact, if $\beta 0,$ even the one-sided inequality $\lambda P(\sup\sb{t\ge 0}Y\sb{t}\ge\lambda)>\beta$ holds. We establish these inequalities by first giving more general inequalities for discrete-time submartingales:$$\lambda P(g\sp\*\ge\lambda)\le 6\Vert f\Vert\sb1$$where $\lambda > 0,\ f = (f\sb{n})\sb{n\ge 0}$ is a submartingle relative to a filtration ${\cal F} = ({\cal F}\sb{n})\sb{n\ge 0}$, and $g = (g\sb{n})\sb{n\ge 0}$ is a process also adapted to ${\cal F}$ that is both differentially and conditionally differentially subordinate to f, i.e. with $f\sb{n} = {\sum\sbsp{k=0}{n}}\ d\sb{k}$ and $g\sb{n} = {\sum\sbsp{k=0}{n}}\ e\sb{k},$ we have that $\vert e\sb{n}\vert\le\vert d\sb{n}\vert$ and $\vert$E$(e\sb{n+1}\vert{\cal F}\sb{n})\vert\le\vert$E$(d\sb{n+1}\vert{\cal F}\sb{n})\vert$ for all $n\ge 0.$ The inequalities obtained are also shown to hold for subharmonic functions and their suitably defined subordinates.
Issue Date:1994
Rights Information:Copyright 1994 Hammack, William
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9503205
OCLC Identifier:(UMI)AAI9503205

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