EXIT TIMES FOR ARMA PROCESSES BY TIMO KOSKI, BRITA JUNG AND GÖRAN HÖGNÄS Abstract We study the asymptotic behaviour of the expected exit time from an interval for the ARMA process, when the noise level approaches 0. Keywords: ARMA model; autoregressive; exit time; first passage time; stationary distribution 2010 Mathematics Subject Classification: Primary 60G17 Secondary 62M10 1. Introduction Autoregressive moving average (ARMA) processes are amongst the most widespread and important tools in time series analysis. In this paper we study exit or first passage problems for these processes and give explicit asymptotic formulae for the expected exit time from an interval for a standard ARMA(n, m) process with normal noise when the noise level approaches 0. The formula is an immediate consequence of results for linear autoregressive (AR) processes in [6]. The essential ingredient in the formula is the invariant probability distribution of an associated multidimensional AR process. This is a centered normal distribution, and hence determined entirely by its covariance matrix. Our work was inspired by a paper on large deviations by Klebaner and Liptser [7]. In an example in that paper, they derived the upper bound of the asymptotic expected exit time of an AR process with normally distributed noise. The corresponding lower bound was then derived in [10] by using Novikov’s martingale method (see [8] and [9]). The result for the asymptotic expected exit time for a multivariate AR process was derived later in [6]. A more general stochastic difference equation of AR type has also been studied, in [4], where an upper bound of the expected exit time was expressed in terms of the invariant probability measure of the process. In this paper we show how the result in [6] can be applied to the ARMA process. Other recent work on related topics can be found for example in [3] and [5], where the probability distribution of the first passage time of an AR(n) process was studied, in [2], which focused on the probability that the AR(n) process does not exceed a barrier before a certain time, and in [1], where an extension of Novikov’s method was used to get a representation of a mean first passage time for the ARMA process, as the mean of an integral containing the process. 2. Exit times for AR processes Jung [6] studied exit times for multivariate AR processes. The methods used also give a result for the expected exit time from an interval of the univariate AR process of order n (the AR(n) process) {X ε t } t ≥0 defined by X ε t = b 1 X ε t −1 +···+ b n X ε t −n + εξ t for t ≥ n, X ε 0 =···= X ε n−1 = 0. (1) doi:10.1017/apr.2018.79 © Applied Probability Trust 2018 191 https://doi.org/10.1017/apr.2018.79 Published online by Cambridge University Press