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.