This paper studies the problem of 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 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 for the habit formation process and on the utility function for the validity of several key assertions of our main results to hold true .
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 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 .
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