This paper considers an initial market model, specified by its underlying assets S and its flow of information \mathbb F, and an arbitrary random time \tau which is not an \mathbb F-stopping time. In this setting, our principal goals reside in describing as explicit as possible the set of all deflators (i.e. the dual set of all wealth processes) and the log-optimal portfolio for the stopped model S^{\tau}. At the practical level, this random time might represent the default time of a firm, the death time of an insured, or more generally an occurrence time of an event that might impact the market somehow. Since the death time and the default time  can be seen when they occur only, the progressive enlargement of \mathbb F with \tau , that we denote by \mathbb G, sounds taylor-fit for modelling the new flow of information that incorporates both \mathbb F and \tau. Thanks to the deep results of Choulli et al (2016) on the ] martingales classification and representation for progressive enlarged filtration, we completely and explicitly describe all the deflators for (S ^{\tau,\mathbb G) in terms of the deflators of the initial market model (S} ,\mathbb F) . This constitute our first principal (and probably the principal) innovative contribution of this paper. Our second principal contribution lies in describing and measuring the impact of \tau on log-related optimal portfolios, namely the num\'eraire portfolio and the portfolio solution to the log-utility maximization problem .
This paper considers an initial market model, specified by its underlying assets S and its flow of information \mathbb F, and an arbitrary random time \tau which might not be an \mathbb F-stopping time. In this setting, our principal goal resides in describing as explicit as possible the set of all deflators , which constitutes the dual set of all "admissible" wealth processes, for the stopped model S^{\tau}. Since the death time and the default time (that \tau might represent) can be seen when they occur only, the progressive enlargement of \mathbb F with \tau sounds tailor-fit for modelling the new flow of information that incorporates both \mathbb F and \tau. Thanks to the deep results of Choulli et al . 8], on martingales classification and representation for progressive enlarged filtration, our aim is fully achieved for both cases of local martingale deflators and general supermartingale delators. The results are illustrated on several particular models for (\tau,S,\mathbb F) such as the discrete-time and the jump-diffusion settings for (S ,\mathbb G) in terms of the deflators of the initial market model (S} ,\mathbb F) , and the case when \tau avoids \mathbb F-stopping times .
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