Controlling the Occupation Time of an Exponential Martingale.

Summary We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between σ 1 and σ 2 , where 0 < σ 1 < σ 2. The optimal control consists in choosing the minimal volatility σ 1 when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold.