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 $(S,\tau )$.