Before revision:  \mbox{%DIFAUXCMD aksamit/choulli/deng/jeanblanc ]  the part after the random time under consideration. We restrict our attention to honest times, and we characterize the pairs of market and honest time for which the resulting model preserves the NUPBR property .
After revision:  \mbox{%DIFAUXCMD aksamit/choulli/deng/jeanblanc ] the part after the occurrence of the random time. Given that all the literature -up to our knowledge- proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property .