 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.
<clarity>  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 , and an arbitrary random time that is not adapted to the flow of the ``public" information.