Terminal wealth maximization under drift uncertainty

We study the portfolio optimization problem of an investor seeking to maximize his terminal wealth. The portfolio is composed of one risky asset, a stock, and one riskless asset, a bond. We assume there is Knightian uncertainty on the stochastic drift term representing the long-term growth rate of the risky asset whose values are not necessarily bounded. We further assume that the investor has a prior estimate about the drift term and quantifies the diffidence of the investor in his prior about the mean. It is assumed that the investor has a logarithmic or power utility. Explicit solutions with unbounded, stochastic and uncertain drift terms have been retrieved. Numerical illustrations revealing the effect of risk awareness and uncertainty on the value function and optimal parameters are also presented.