An Explicit Solution for an Optimal Stopping/optimal Control Problem Which Models an Asset Sale

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function. The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the set-up is based on continuous diffusions, the optimal martingale may involve a jump process. One interpretation of the solution is that it is optimal for the risk averse agent to gamble.