Optimal Portfolio and Consumption in a Switching Diffusion Market

Abstract

This paper addresses the problem of finding the optimal portfolio and consumption of a small agent in an economy. The novelty of this work is in considering that the financial market, in contrast to the celebrated Black-Scholes model, is composed of two sources of uncertainties: a Brownian motion and a continuous time Markov chain. While the Brownian motion intends to model the normal oscillations of the asset prices, the continuous time Markov chain aims at taking into account the abrupt variations that can occur in the parameters of the asset dynamics due to changes that take place in the state of the economy. The problem is formulated in terms of classical optimal stochastic control and the Hamilton-Jacobi-Bellman equation is solved to yield the solution.