  In this paper, we consider an informational market model with two flows of
informations. The smallest flow F, which is available to all agents, is the
filtration of the initial market model(S,F,P), where S is the assets' prices
and P is a probability measure. The largest flow G contains additional
information about the occurrence of a random time T. This setting covers credit
risk theory where T models the default time of a firm, and life insurance where
T represents the death time of an insured. For the model (S-S^T,G,P), we
address the log-optimal portfolio problem in many aspects. In particular, we
answer the following questions and beyond: 1) What are the necessary and
sufficient conditions for the existence of log-optimal portfolio of the model
under consideration? 2) what are the various type of risks induced by T that
affect this portfolio and how? 3) What are the factors that completely describe
the sensitivity of the log-portfolio to the parameters of T? The answers to
these questions and other related discussions definitely complement the work of
Choulli and Yansori [12] which deals with the stopped model (S^T,G).
