Before revision:  With the path dependence reduction and the embedding approach, the existence and uniqueness of the optimal consumption are proved using the auxiliary primal and dual processes and the convex duality analysis .
After revision:  With the path dependence reduction and the embedding approach, the existence and uniqueness of the optimal consumption are proved using the auxiliary primal and dual processes and the duality approach. 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 habits impact, however, in a modified isomorphic market model .