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 On $\Phi$-inequalities for bounded submartingales and subharmonic functions Commun. Korean Math. Soc. 2008 Vol. 23, No. 2, 269-277 Printed June 1, 2008 Adam Os\c ekowski Warsaw University Abstract : Let $f=(f_n)$ be a nonnegative submartingale such that $||f||_\infty \leq 1$ and $g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $$|dg_n| \leq |df_n|,\ \ \ n=0,\,1,\,2,\,\ldots.$$ The paper contains the proof of the sharp inequality $$\sup_n \E \Phi(|g_n|) \leq \Phi(1)$$ for a class of convex increasing functions $\Phi$ on $[0,\infty)$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain. Keywords : martingale, submartingale, stochastic integral, $\Phi$-inequality, differential subordination MSC numbers : Primary 60G42; Secondary 60H05 Downloads: Full-text PDF

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