Before revision:  For the  stopped model (S^{\tau},\mathbb G), we describe in different manners and as explicit as possible the num\'eraire portfolio, the log-optimal portfolio, the log-optimal deflator (which is the dual of the log-optimal portfolio), and we elaborate their duality without any further assumption .
After revision:  For the resulting stopped model (S^{\tau},\mathbb G), we describe in different manners and as explicit as possible the num\'eraire portfolio, the log-optimal portfolio, the log-optimal deflator (which is the dual of the log-optimal portfolio), and we elaborate their duality without any further assumption .