This paper considers an initial market model, specified by the pair $(S,\mathbb F)$ where $S$ is its discounted assets' price process and $\mathbb F$ its flow of information, and an arbitrary random time $\tau$. This random time can represent the death time of an agent or the default time of a firm, and in both cases $\tau$ can not be seen before it occurs. Thus, the progressive enlargement of $\mathbb F$ with $\tau$, denoted by $\mathbb G$, sounds tailor-fit for modelling the new flow of information that incorporates both $\mathbb F$ and $\tau$. 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.
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