This paper studies continuous time utility maximization of consumption together with addictive habit formation in general incomplete semimartingale financial markets. By introducing the auxiliary state processes and the modified dual space, we embed our original problem into an auxiliary time separable utility maximization problem with the shadow random endowment. We establish existence and uniqueness of the optimal solution using convex duality approach on the product space _{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} by defining the primal value function both on the initial wealth and initial habit . We also provide market independent sufficient conditions both on stochastic discounting processes of the habit formation process and on the utility function  to modify the convex duality approach when the auxiliary dual process is not necessarily integrable.
This paper studies the problem of continuous time expected utility maximization of consumption together with addictive habit formation in general incomplete semimartingale financial markets. Introducing an auxiliary state processes and a modified dual space, we embed our original problem into an auxiliary time-separable utility maximization problem with the shadow random endowment. We establish existence and uniqueness of the optimal solution using convex duality on the product space \mathbb{L_{+}^{0}(\Omega\times[0,T],O,}\mathbb{P}) by defining the primal value function as depending on both the initial wealth and initial standard of living . We also provide market independent sufficient conditions on both stochastic discounting processes of the habit formation process and on the utility function for our original problem to be well posed and to modify the convex duality approach when the auxiliary dual process is not necessarily integrable.
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