  This paper studies the optimal consumption under the addictive habit
formation preference in markets with transaction costs and unbounded random
endowments. To model the proportional transaction costs, we adopt the Kabanov's
multi-asset framework with a cash account. At the terminal time T, the investor
can receive unbounded random endowments for which we propose a new definition
of acceptable portfolios based on the strictly consistent price system (SCPS).
We prove a type of super-hedging theorem using the acceptable portfolios which
enables us to obtain the consumption budget constraint condition under market
frictions. Applying the path dependence reduction and the embedding approach,
we obtain the existence and uniqueness of the optimal consumption using some
auxiliary processes and the duality analysis. As an application of the duality
theory, the market isomorphism with special discounting factors is also
discussed in the sense that the original optimal consumption with habit
formation is equivalent to the standard optimal consumption problem without the
habits impact, however, in a modified isomorphic market model.
