This paper studies the problem of continuous time expected utility maximization of consumption together with addictive habit formation in general incomplete semimartingale markets. Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into an abstract time-separable utility maximization problem with a shadow random endowment on the product space . We establish existence_{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} and uniqueness of the optimal solution  using convex duality by defining the primal value function as depending on two variables, i.e., the initial wealth and the initial standard of living. We also provide market independent sufficient conditions both on the stochastic discounting processes and on the utility function for the well-posedness of our original optimization problem. Under the same assumptions, we can carefully modify the classical proofs in the approach of convex duality analysis  when the auxiliary dual process is not necessarily integrable.
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],O,}\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, i.e., 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.
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