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 .
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) .
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considers an initial market model , specified by the pair
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
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where S is its discounted
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, and an arbitrary
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setting covers the areas of credit risk and life insurance, where \tau represents
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of a firm, and in both cases \tau can not be seen before it occurs
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resulting
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describe
study the two portfolios
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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
, and describe their computations in terms of the \mathbb F-observable parameters of the pair (S, \tau)