OPTIMAL PORTFOLIOS WITH STOCHASTIC SHORT RATE: PITFALLS WHEN THE SHORT RATE IS NON-GAUSSIAN OR THE MARKET PRICE OF RISK IS UNBOUNDED

The aim of this paper is to provide a survey of some of the problems occurring in portfolio problems with power utility, Non-Gaussian interest rates, and/or unbounded market price of risk. Using stochastic control theory, we solve several portfolio problems for different specifications of the short rate and the market price of risk. In particular, we consider a Gaussian model, the Cox-Ingersoll-Ross model, and squared Gaussian as well as lognormal specifications of the short rate. We find that even in a Gaussian framework the canonical candidate for the value function may not be finite if the market price of risk is unbounded. It is thus not straightforward to generalize results on continuous-time portfolio problems with power utility, Gaussian interest rates, and bounded market price of risk to situations where the short rate is Non-Gaussian or the market price of risk is unbounded.