  This paper focuses on num\'eraire portfolio and log-optimal portfolio
(portfolio with finite expected utility that maximizes the expected logarithm
utility from terminal wealth), when a market model $(S,\mathbb F)$ -specified
by its assets' price $S$ and its flow of information $\mathbb F$- is stopped at
a random time $\tau$. This setting covers the areas of credit risk and life
insurance, where $\tau$ represents the default time and the death time
respectively. 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 resulting
stopped model $(S^{\tau},\mathbb G)$, we study the two portfolios in different
manners, and describe their computations in terms of the $\mathbb F$-observable
parameters of the pair $(S, \tau)$.
