Before revision: 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.
After revision: 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) where S is its discounted assets' price process and \mathbb F its flow of information , and an arbitrary random time \tau.