  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 setup 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.
