An optimal investment model with Markov-driven volatilities

Abstract

We consider a multi-stock market model. The processes of stock prices are governed by stochastic differential equations with stock return rates and volatilities driven by a finite-state Markov process. Each volatility is also disturbed by a Brownian motion; more exactly, it follows a Markov-driven Ornstein–Uhlenbeck process. Investors can observe the stock prices only. Both the underlying Brownian motion and the Markov process are unobservable. We study a discretized version, which is a discrete-time hidden Markov process. The objective is to control trading at each time step to maximize an expected utility function of terminal wealth. Exploiting dynamic programming techniques, we derive an approximate optimal trading strategy that results in an expected utility function close to the optimal value function. Necessary filtering and forecasting techniques are developed to compute the near-optimal trading strategy.