Extremes of Locally Brownian Processes

Abstract

This chapter looks at extrema and boundary crossings for stationary and near-stationary 1-dimensional processes which are “locally Brownian”. The prototype example is the Ornstein-Uhlenbeck process, which is both Gaussian and Markov. One can then generalize to non-Gaussian Markov processes (diffusions) and to non-Markov Gaussian processes; and then to more complicated processes for which these serve as approximations. In a different direction, the Ornstein-Uhlenbeck process is a time and space-change of Brownian motion, so that boundary-crossing problems for the latter can be transformed to problems for the former: this is the best way to study issues related to the law of the iterated logarithm.