Rare-event analysis of modulated Ornstein–Uhlenbeck processes

Abstract

ABSTRACT. This paper studies Ornstein-Uhlenbeck (OU) processes in a random environment. The OU model has found widespread use in networking, as a Gaussian approximation of the user-level dynamics that allows explicit analysis; adding modulation to it allows incorporating phenomena in which the users' activity level is affected by exogenous factors. The focus lies on rare-event analysis: under a specific scaling of the parameters involved, we establish the large deviations asymptotics of the probability that the process reaches an extreme value. The decay rate of this probability is generally only implicitly available (as the solution to a variational problem), but specializing to the case of Markov modulation we succeed in devising efficient numerical procedures.