Semimartingales and the standard Brownian motion (Q1120194)

Let \((b_ t\), \(t\geq 0)\) be a standard Brownian motion. The purpose of this note is to characterize all functions \(\phi\) such that \((t\phi (b_ t)\), \(t\geq 0)\) is a submartingale (resp. a semimartingale). Such a function \(\phi\) turns out to be a non-negative convex function (resp. a difference of two convex functions). These results are closely connected with those of \textit{E. Cinlar}, \textit{J. Jacod}, \textit{P. Protter} and \textit{M. J. Sharpe} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 161-219 (1980; Zbl 0443.60074)], and they are used here to give geometrical interpretations of certain results in the paper of the author and \textit{Nguyen Xuan Loc}, Characterization of functions which transform Brownian sheet into planar semimartingale. Preprint (1985).