Optimal Consumption under Habit Formation In Markets with Transaction Costs and Random Endowments

TL;DR

This paper develops a comprehensive framework for optimal consumption with habit formation in markets featuring transaction costs and random endowments, establishing existence, uniqueness, and market isomorphism results.

Contribution

It introduces a new acceptable portfolio concept based on SCPS, proving super-hedging theorems and establishing duality and market isomorphism in complex market settings.

Findings

Existence and uniqueness of optimal consumption are proven.

A new super-hedging theorem using acceptable portfolios is established.

Market isomorphism with modified discounting factors is demonstrated.

Abstract

This paper studies the optimal consumption under the addictive habit formation preference in markets with transaction costs and unbounded random endowments. To model the proportional transaction costs, we adopt the Kabanov's multi-asset framework with a cash account. At the terminal time T, the investor can receive unbounded random endowments for which we propose a new definition of acceptable portfolios based on the strictly consistent price system (SCPS). We prove a type of super-hedging theorem using the acceptable portfolios which enables us to obtain the consumption budget constraint condition under market frictions. Applying the path dependence reduction and the embedding approach, we obtain the existence and uniqueness of the optimal consumption using some auxiliary processes and the duality analysis. As an application of the duality theory, the market isomorphism with special discounting factors is also discussed in the sense that the original optimal consumption with habit formation is equivalent to the standard optimal consumption problem without the habits impact, however, in a modified isomorphic market model.