 In this paper, we consider a semi-martingale market model (initial market model) fulfilling these structures , and an arbitrary random time that is not adapted to the  flow of the ``public" information.
<meaning-changed>  In this paper, we consider a semi-martingale market model (initial market model) fulfilling these structures , and an arbitrary random time that is not adapted to the  flow of the ``public" information.
 In this paper, we consider a semi-martingale market model (initial market model) fulfilling these structures , and an arbitrary random time that is not adapted to the flow of the market model. This random time may model the default time of a firm, the death time of an insured, or any the occurrence time of an event that might impact the market model somehow.