# American Institute of Mathematical Sciences

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December  2016, 6(4): 629-651. doi: 10.3934/mcrf.2016018

## An infinite time horizon portfolio optimization model with delays

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-7913, United States 2 Operations Research, North Carolina State University, Raleigh, NC 27695-7913, USA; Current address: Department of Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan

Received  October 2015 Revised  January 2016 Published  October 2016

In this paper we consider a portfolio optimization problem of the Merton's type over an infinite time horizon. Unlike the classical Markov model, we consider a system with delays. The problem is formulated as a stochastic control problem on an infinite time horizon and the state evolves according to a process governed by a stochastic process with delay. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton-Jacobi-Bellman (HJB) equations in a finite dimensional space for logarithmic and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are derived, too.
Citation: Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control & Related Fields, 2016, 6 (4) : 629-651. doi: 10.3934/mcrf.2016018
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