Optimal Investment and Consumption Problem with Stochastic Environments

Abstract

Optimal investment and consumption problem for a CRRA investor or agent is solved in this study. An agent invests in the financial market with one risk-free security and one risky security. The stochastic interest rate dynamics of risk-free security follow a Ho-Lee model and the risky security is modeled as Heston’s model with its volatility parameter dynamics following a Cox-Ingersoll-Ross (CIR) model. Interest rates and volatility rates, in reality, are stochastic due to uncertain events such as the Coronavirus disease 2019 (COVID19) pandemic, climate change, etc. Our main goal is to allocate initial wealth x0 between risk-free security and risky security in order to maximize the discounted expected utility of consumption and terminal wealth over a finite horizon. Applying the Dynamic Programming Principle (DPP), the HJB PDE for the value function is established. The power utility function which belongs to the Constant Relative Risk Aversion (CRRA) class is employed for our analysis to obtain the value function and optimal policies. Finally, numerical examples and simulations are provided and discussed.