An optimal consumption-investment model with constraint on consumption

Zuo Quan Xu; Fahuai  Yi

doi:10.3934/mcrf.2016014

Article Contents

2016, Volume 6, Issue 3: 517-534. Doi: 10.3934/mcrf.2016014

This issue Previous Article A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations Next Article

An optimal consumption-investment model with constraint on consumption

Zuo Quan Xu^1, and
Fahuai Yi^2,

1.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
2.
School of Finance, Guangdong University of Foreign Studies, Guangzhou 510420

Received: May 31, 2015

Revised: February 29, 2016

Published: August 2016

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Abstract

A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize the total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique (known) possibly exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth. According to this model, an investor should take the similar investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.

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Mathematics Subject Classification: Primary: 91G10; Secondary: 91G80, 49L25.

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