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OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY

Part of: Mathematical finance

Published online by Cambridge University Press:  18 February 2019

L. LI  and
H. MI Open the ORCID record for H. MI [Opens in a new window]
L. LI
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, China email perfectlyli@163.com, mihui@njnu.edu.cn
H. MI*
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, China email perfectlyli@163.com, mihui@njnu.edu.cn
*
✉mihui@njnu.edu.cn
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Abstract

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We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.

Keywords

stochastic differential delay equationpower utility functionstochastic factorCox–Ingersoll–Ross model

MSC classification

Primary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 91G10: Portfolio theory
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 1 , January 2019 , pp. 99 - 117
Copyright
© 2019 Australian Mathematical Society 

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