Before revision:  The second type of results consists of giving necessary and sufficient conditions on the random time for which the non-arbitrage is preserved for any process. Our class of honest times ---that we consider in the paper--- is much larger than the class of all stopping times, and plays an important r\^ole in classifying random times. 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 second type of results consists of giving necessary and sufficient conditions on the random time  optional stochastic integral (or compensated stochastic integral ) that was introduced in early eighties .