Optimal Investment Strategy for Merton´s Portfolio Optimization Problem under a CEV Model

This paper studies Merton´s portfolio problem for an investor who can trade a risk-free asset and a stock. We consider the case where the stock price follows a constant elasticity of variance (CEV) dynamics. By applying stochastic optimal control theory and variable change technique, we derive an explicit solution for the CRRA utility function. The solution consists of a moving Merton strategy and a modified factor. The moving Merton strategy is equivalent to the original Merton strategy but it has an updated volatility at the current time. The modified factor denotes a supplement part resulted from the change of the volatility.