Valeurs Prises par les Martingales Locales Positives Continues a un Instant Donne

Abstract

In a previous paper we proved that a necessary and sufficient condition for all martingales of a given filtration $(\mathscr{F}_t)$ to be continuous is that, for every stopping time $T$ and every $\mathscr{F}_T$-measurable random variable $X$, there exists a continuous local martingale $M$ with $M_T = X$ a.s. The aim of this paper is to study the following question: Can we choose $M \geq 0$ whenever $X \geq 0$? We also give a negative answer to Conjecture 7.1 of Harrison and Pliska.