The paper studies optimal portfolio selection for discrete timemarket models in mean-variance and goal achieving setting. Theoptimal strategies are obtained for models with an observed processthat causes serial correlations of price changes. The optimalstrategies are found to be myopic for the goal-achieving problem andquasi-myopic for the mean variance portfolio.

One of two random variables, X and Y, can be selected at each of a possibly infinite number of stages. Depending on the outcome, one's fortune is either increased or decreased by 1. The probability of increase may not be known for either X or Y. The objective is to increase one's fortune to G before it decreases to g, for some integral g and G; either may be infinite.

In Part I (Berry and Fristedt (1980)), the distribution of X is unknown and that of Y is known. In the current part, it is known that either X or Y has probability α of increasing the current fortune by 1 and the other has probability β of increasing the fortune by 1, where α and β are known, but which goes with X is not known. We show that optimal strategies exist in general and find all optimal schemes when α = 0 and when α + β = 1. In both cases myopic strategies are shown to be optimal. A counterexample is used to show that myopic strategies, while intuitively very appealing, are not optimal for general (α, β).