Minimizing the lifetime shortfall or shortfall at death

In this paper, we introduce the other three criteria in order to reduce the leveraging observed by Young, the above mentioned reference. We discovered that surprisingly the optimal investment strategy for criterion (3) is identical to the one for (1) and that the strategies for (2) and (4) are more leveraged than the one for (1) at low wealth. Because these criteria do not reduce leveraging, we completely remove it by considering problems (1) and (3) under the restriction that the individual cannot borrow to invest in the risky asset.