A consumption-investment model with wealth-dependent lower bound constraint on consumption

Chonghu Guan, Zuo Quan Xu, Fahuai Yi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

This paper studies a life-time consumption-investment problem under the BlackScholes framework, where the consumption rate is subject to a lower bound constraint that linearly depends on her wealth. It is a stochastic control problem with state-dependent control constraint to which the standard stochastic control theory cannot be directly applied. We overcome this by transforming it into an equivalent stochastic control problem in which the control constraint is state-independent so that the standard theory can be applied. We give an explicit optimal consumptioninvestment strategy when the constraint is homogeneous. When the constraint is non-homogeneous, it is shown that the value function is third-order continuously differentiable by differential equation approach, and a feedback form optimal consumption-investment strategy is provided. According to our findings, if one is concerned with long-term more than short-term consumption, then she should always consume as few as possible; otherwise, she should consume optimally when her wealth is above a threshold, and consume as few as possible when her wealth is below the threshold.
Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalJournal of Mathematical Analysis and Applications
Publication statusE-pub ahead of print - Dec 2021

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