Optimal singular control strategies for controlling a process to a goal

An investor starting with initial wealth z0>0 would like to achieve a total wealth a where a>z0 before going bankrupt. The strategy is to allocate his wealth between a chosen risky asset and a bank account. The amount invested in the risky asset is given by an Itô process with infinitesimal parameters [mu] and [sigma]. At time t, the choice of the risky asset is represented by [mu](t) and [sigma](t), which comes from a control set. This control set depends on the investor's wealth in the risky asset. At any time wealth can be transferred between the risky asset and the bank account without any transaction fee as long as the transaction process is of bounded variation. The problem considered here is to find an optimal strategy, which consists of an optimal choice of a risky asset and allocation of wealth, to maximize the probability of reaching a total wealth of a.