  This paper studies the continuous time utility maximization problem on
consumption with addictive habit formation in incomplete semimartingale
markets. Introducing the set of auxiliary state processes and the modified dual
space, we embed our original problem into a time-separable utility maximization
problem with a shadow random endowment on the product space
$\mathbb{L}_+^0(\Omega\times [0,T],\mathcal{O},\overline{\mathbb{P}})$.
Existence and uniqueness of the optimal solution are established using convex
duality approach, where the primal value function is defined on two variables,
that is, the initial wealth and the initial standard of living. We also provide
sufficient conditions on the stochastic discounting processes and on the
utility function for the well-posedness of the original optimization problem.
Under the same assumptions, classical proofs in the approach of convex duality
analysis can be modified when the auxiliary dual process is not necessarily
integrable.
