The Annals of Applied Probability

Utility maximization with addictive consumption habit formation in incomplete semimartingale markets

Xiang Yu

Full-text: Open access

Abstract

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 $\mathbb{L}_{+}^{0}(\Omega\times[0,T],\mathcal{O},\overline{\mathbb{P}})$. Existence and uniqueness of the optimal solution are established using convex duality approach, where the primal value function is defined on two variables, that is, 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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1383-1419.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124132

Digital Object Identifier
doi:10.1214/14-AAP1026

Mathematical Reviews number (MathSciNet)
MR3325277

Zentralblatt MATH identifier
1312.91084

Subjects
Primary: 91G10: Portfolio theory 91B42: Consumer behavior, demand theory
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Time nonseparable utility maximization consumption habit formation auxiliary processes convex duality incomplete markets

Citation

Yu, Xiang. Utility maximization with addictive consumption habit formation in incomplete semimartingale markets. Ann. Appl. Probab. 25 (2015), no. 3, 1383--1419. doi:10.1214/14-AAP1026. https://projecteuclid.org/euclid.aoap/1427124132


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