On a class of the universally integrable random functions

Abstract

([0, 1]).The purpose of this paper is to show that if we replace the usualIto integral by the extended Ito integral (E.I.I., for short) in the defini-tion of the quasi-martingale (1), we also obtain an analogous theorem.For the reader's convenience, we recall the basic facts concerning theE.I.I, following [6], which gives a rather systematic survey of recentdevelopments in this theory, mainly due to the Soviet school (such asDaletskii-Paramonova, Shevljakov, Skorokhod etc.).Let