An infinite time horizon portfolio optimization model  with delays

Tao  Pang; Azmat  Hussain

doi:10.3934/mcrf.2016018

Article Contents

2016, Volume 6, Issue 4: 629-651. Doi: 10.3934/mcrf.2016018

This issue Previous Article Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions Next Article Forward-backward evolution equations and applications

An infinite time horizon portfolio optimization model with delays

Tao Pang^1, and
Azmat Hussain^2,

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

Received: September 30, 2015

Revised: December 31, 2015

Published: November 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 91G80, 49L20; secondary: 91G10, 93E20.

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