Optimal Portfolios with Bounded Capital at Risk

We consider some continuous‐time Markowitz type portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the capital at risk. In a Black–Scholes setting we obtain closed‐form explicit solutions and compare their form and implications to those of the classical continuous‐time mean‐variance problem. We also consider more general price processes that allow for larger fluctuations in the returns.