A Note on Maximal Inequality for Stochastic Convolutions

Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int_0^t {S(t - s)} \psi (s){\text{d}}W(s)$$ \end{document} driven by a Wiener process W in a Hilbert space in the case when the semigroup S(t) is of contraction type.