  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 known in the literature as "{\it Structure Conditions}". In this
paper, we consider a semi-martingale market model, and an arbitrary random time
that is not adapted to the information 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. By
adding additional uncertainty to the market model, via this random time, the
{\it structures conditions} may fail and hence the Markowitz's optimal
portfolio and other quadratic-optimal portfolios might fail to exist. Our aim
is to investigate the impact of this random time on the structures conditions
from different perspectives. Our analysis allows us to conclude that under some
mild assumptions on the market model and the random time, these structures
conditions 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 the random time and incorporating totally a
specific class of random times respectively.
