Optimal Consumption With Habit Formation In Markets with Transaction Costs And Unbounded Random Endowment

This paper studies the utility maximization problem on consumption with addictive habit formation in the market with proportional transaction costs and unbounded random endowment. To model the proportional transaction costs, we adopt the Kabanov's multi-asset framework with a cash account. At the terminal time $t=T$, the investor can receive an unbounded random endowment for which we propose a new definition of acceptable portfolio processes depending on the strictly consistent price system (SCPS). We prove a type of super-hedging theorem for a family of workable contingent claims using the acceptable portfolios and random endowment which enables us to obtain the consumption budget constraint result under the market frictions. With the path dependence reduction and the embedding approach, the existence and uniqueness of the optimal consumption are proved using the auxiliary primal and dual processes and the convex duality analysis.