Optimal stopping of Brownian motion with broken drift

Abstract

AbstractWe solve an optimal stopping problem where the underlying diffusion is Brownian motion on with a positive piecewise constant drift changing at zero. It is assumed that the drift μ1 on the negative side is smaller than the drift μ2 on the positive side. The main observation is that if , then there exist values of the discounting parameter for which it is not optimal to stop in the vicinity of zero where the drift changes. However, when the discounting gets larger, the stopping region becomes connected and contains zero. This is in contrast with results concerning optimal stopping of skew Brownian motion where the skew point is for all values of the discounting parameter in the continuation region in case the skew parameter is bigger than 1/2. In all cases, a practical implementation of these optimal stopping strategies, based on hitting times of boundaries of continuation regions, would require access to the highest‐possible frequency data which is consistent with the continuous‐time modeling framework.