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Winter 2011 A sharp weak-type bound for Itō processes and subharmonic functions
Adam Osȩkowski
Kyoto J. Math. 51(4): 875-890 (Winter 2011). DOI: 10.1215/21562261-1424893


Let α0, and let X, Y be Itō processes

dXt=ϕtdBt+ψtdt, dYt=ζtdBt+ξtdt

such that |X0||Y0|, |ϕ||ζ|, and αψ|ξ|. The purpose of the paper is to determine the optimal universal constant Cα in the weak-type estimate

sup λλP(sup t|Yt|λ)Cαsup tE|Xt|.

Then the inequality is extended, with unchanged constant, to the more general setting when X is a submartingale and Y is α-strongly differentially subordinate to X. As an application, a related estimate for subharmonic functions is established. The inequalities generalize and unify the earlier results of Burkholder, Choi, and Hammack for Itō processes, stochastic integrals, and smooth functions on Euclidean domains.


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Adam Osȩkowski. "A sharp weak-type bound for Itō processes and subharmonic functions." Kyoto J. Math. 51 (4) 875 - 890, Winter 2011.


Published: Winter 2011
First available in Project Euclid: 10 November 2011

MathSciNet: MR2854156
Digital Object Identifier: 10.1215/21562261-1424893

Primary: 60G44
Secondary: 31A05 , 60H05

Rights: Copyright © 2011 Kyoto University

Vol.51 • No. 4 • Winter 2011
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