Log-optimal portfolio and num\'eraire portfolio for market models stopped at a random time

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 $(S, \tau)$.