Utility Maximization with Addictive Consumption Habit Formation in Incomplete Semimartingale Markets

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 $\mathbb{L}_{+}^{0}(\Omega\times[0,T],\mathcal{O},\bar{\mathbb{P}})$. 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.