Choosing an investment strategy by stochastic control

A portfolio optimisation problem on an infinite time horizon is considered. Risky asset price obeys a logarithmic Brownian motion, and the interest rate varies according to a Markov diffusion process. This paper obtains an investment strategy considering one stock, one bond where the risk-free interest rate, the appreciation and the volatility of the stock depend on an external finite state Markov chain. We investigate the problem of maximising the expected utility from terminal wealth and solve it explicitly by stochastic control methods for a specific utility function U (x ) = logx.