Before revision:  The crucial stochastic tool that drives our analysis lies in the optional stochastic integral (or compensated stochastic integral ) that was introduced in early eighties .
After revision:  The crucial stochastic tool that drives our analysis lies in the optional stochastic integral (or compensated stochastic integral models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time .