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,}%DIFDELCMD < \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.
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 and uniqueness of the optimal solution using convex duality _{+}^{0}(\Omega\times[0,T],O,}%DIFDELCMD < \mathbb{P}%%% 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.
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