  This paper addresses the log-optimal portfolio for a general semimartingale
model. The most advanced literature on the topic elaborates existence and
characterization of this portfolio under no-free-lunch-with-vanishing-risk
assumption (NFLVR). There are many financial models violating NFLVR, while
admitting the log-optimal portfolio on the one hand. On the other hand, for
financial markets under progressively enlargement of filtration, NFLVR remains
completely an open issue, and hence the literature can be applied to these
models. Herein, we provide a complete characterization of log-optimal portfolio
and its associated optimal deflator, necessary and sufficient conditions for
their existence, and we elaborate their duality as well without NFLVR.
