Optimal investment problem with complete memory on an infinite time horizon

Abstract

This paper deals with an optimal investment problem with complete memory over an infinite time horizon, where the appreciation rate of the risky asset depends on the historical information and the wealth process is governed by a stochastic delay differential equation. The investor’s objective is to maximize his expected discounted utility for the combination of the terminal wealth and the average performance by choosing optimal investment and consumption as controls. Under some suitable conditions, we provide the corresponding Hamilton-Jacobin-Bellman(HJB) equation, and derive the explicit expressions of the optimal investment and consumption for exponential utility function. The verification theorem is also established. Moreover, some numerical examples and sensitivity analysis are given to illustrate our results.