Composition of an efficient portfolio in the Bielecki and Pliska market model

We study a continuous time portfolio optimization model due to Bielecki and Pliska where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors. We introduce a functional $Q_\gamma$ that features the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $\gamma$ to the variance when we keep the value of the factor levels fixed. The coefficient $\gamma$ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for $Q_\gamma$ at any moment of time. Single-factor case is analyzed in more details. We give a simple asset allocation example featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of risk-sensitive control theory thereby using an infinite horizon objective that features the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to $\gamma.$