ON Φ-INEQUALITIES FOR BOUNDED SUBMARTINGALES AND SUBHARMONIC FUNCTIONS

Title & Authors
ON Φ-INEQUALITIES FOR BOUNDED SUBMARTINGALES AND SUBHARMONIC FUNCTIONS

Abstract
Let $\small{f=(f_n)}$ be a nonnegative submartingale such that $\small{{\parallel}f{\parallel}{\infty}{\leq}1\;and\;g=(g_n)}$ be a martingale, adapted to the same filtration, satisfying $\small{{\mid}d_{gn}{\mid}{\leq}{\mid}df_n{\mid},\;n=0,\;1,\;2,\;....}$ The paper contains the proof of the sharp inequality $\small{\limits^{sup}_ n\;\mathbb{E}{\Phi}({\mid}g_n{\mid}){\leq}{\Phi}(1)}$ for a class of convex increasing functions $\small{{\Phi}\;on\;[0,\;{\infty}]}$, 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$\small{{\Phi}}$-inequality;differential subordination;
Language
English
Cited by
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