Controlling the Occupation Time of an Exponential Martingale

Abstract

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 \(\sigma _1\) and \(\sigma _2\), where \(0 < \sigma _1 < \sigma _2\). The optimal control consists in choosing the minimal volatility \(\sigma _1\) 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.