Reaching goals under ambiguity: Continuous-time optimal portfolio selection

In this paper, the problem of reaching goals for an investor in the financial market is studied. We follow the context of Jin and Zhou (2015) who studied the continuous-time optimal portfolio selection in which the appreciation rates are only known to be in a certain convex closed set. The cost functional in our problem is of type I{x≥1} which is not concave or convex, even not continuous, we prove that the min-max theorem is still applicable. Via PDE approach, we construct the optimal portfolio strategy. At last, we obtain the saddle point for our problem explicitly.