An explicit solution for an optimal stopping/optimal control problem which models an asset sale

摘要

In this article we study an optimal stopping/optimal control problem whichmodels the decision facing a risk-averse agent over when to sell an asset. Themarket is incomplete so that the asset exposure cannot be hedged. In additionto the decision over when to sell, the agent has to choose a control strategywhich corresponds to a feasible wealth process. We formulate this problem asone involving the choice of a stopping time and a martingale. We conjecture theform of the solution and verify that the candidate solution is equal to thevalue function. The interesting features of the solution are that it isavailable in a very explicit form, that for some parameter values the optimalstrategy is more sophisticated than might originally be expected, and thatalthough the setup is based on continuous diffusions, the optimal martingalemay involve a jump process. One interpretation of the solution is that it isoptimal for the risk-averse agent to gamble.