Before revision:  Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a specific honest time. For the general case of semimartingale, we provide necessary and sufficient conditions for the NUPBR to be preserved. Precisely, we elaborate two types of results. For a fixed semimartingale and a random time, we provide necessary and sufficient conditions on the process and the random time for which the non-arbitrage concept remains valid .
After revision:  Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a specific For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the non-arbitrage concept remains valid .