It has been understood that the ``local" existence of the Markowitz' optimal portfolio or the solution to the local risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets price processes (called ``Structure Conditions" in the literature). 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. By adding additional uncertainty to the initial market model, via this random time, those structures may fail. Our aim is to address the question of how this random time will affect these structures from different perspectives. Our analysis allowed us to conclude that under some mild assumptions on the market model and the random time, these structures will remain valid on the one hand. Furthermore, we provide two examples illustrating the importance of these assumptions. On the other hand, we describe the random time models for which these structure conditions are preserved for any market model. These results are elaborated separately for the two contexts of stopping with random time and incorporating totally a specific class of random times respectively.
