Distributional and Extremal Behaviour of Brownian Motion with Exponential Resetting

We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the infimum. We also consider the stationary case and give a new explicit expression for the fidi’s of such processes.