A note on exit times for nonlinear autoregressive processes

Abstract

We study the exit times from a bounded interval for a nonlinear autoregressive process of order one, denoted by X(f) := {X n (f), n = 1,2,...} where the process is defined by the recurrence relation (1) with a continuous, contractive function f : R 1 → R 1 , a small positive noise parameter ε > 0, and a sequence {ξ n } of independent and identically distributed standard normal random variables. Klebaner and Liptser (see [1;2]) applied the large deviation principle to obtain key asymptotic estimates for the exit times from the interval [−1,1] for linear AR(1) processes. Building on their results, G. Hognas and B. Jung [3] derived upper bounds for exit times in the case of AR(1) processes driven by several piecewise-linear maps on [−1,1]. In the present work, we extend the results of Hognas and Jung by considering a broader class of piecewise continuous maps f. We show that, for this class, the asymptotic behavior of the exit times depends critically on both the slopes and the locations of the breakpoints of f