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 $(S,\mathbb F)$, and the case when $\tau$ avoids $\mathbb F$-stopping times.
↧