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An optimal consumption and investment problem with partial information

Part of: Stochastic systems and control Mathematical finance Hamilton-Jacobi theories, including dynamic programming Stochastic analysis

Published online by Cambridge University Press:  20 March 2018

Hiroaki Hata  and
Shuenn-Jyi Sheu
Hiroaki Hata*
Affiliation:
Shizuoka University
Shuenn-Jyi Sheu*
Affiliation:
National Central University
*
* Postal address: Department of Mathematics, Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422-852, Japan. Email address: hata@shizuoka.ac.jp
** Postal address: Department of Mathematics, National Central University, Chung-Li, 320-54, Taiwan. Email address: sheusj@math.ncu.edu.tw
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Abstract

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We consider a finite-time optimal consumption problem where an investor wants to maximize the expected hyperbolic absolute risk aversion utility of consumption and terminal wealth. We treat a stochastic factor model in which the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case in which the investor cannot observe the factor process and uses only past information of risky assets. Then our problem is formulated as a stochastic control problem with partial information. We derive the Hamilton–Jacobi–Bellman equation. We solve this equation to obtain an explicit form of the value function and the optimal strategy for this problem. Moreover, we also introduce the results obtained by the martingale method.

Keywords

Optimal consumption and investmentHARA utilitystochastic factor modelpartial informationHamilton–Jacobi–Bellman equation

MSC classification

Primary: 49L20: Dynamic programming method 60H30: Applications of stochastic analysis (to PDE, etc.) 91G10: Portfolio theory 93E20: Optimal stochastic control
Secondary: 93E11: Filtering
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 131 - 153
Copyright
Copyright © Applied Probability Trust 2018 

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