Maximal weak-type inequality for stochastic integrals

Adam Osekowski (University of Warsaw)


Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates.  (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$
(ii) If $X$ is arbitrary, then$$  \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.

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Pages: 1-13

Publication Date: May 4, 2014

DOI: 10.1214/ECP.v19-3151


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