Small-noise asymptotics of exit times for a class of nonlinear autoregressive processes

Abstract

We study exit times from the interval $[-1,1]$ for the nonlinear autoregressive process given, for $n\geq 0$, by $$ X^{(\varepsilon)}_{n+1}(\varphi_{f,g}) =\varphi_{f,g}\bigl(X^{(\varepsilon)}_{n}\bigr)+\varepsilon\xi_{n+1}, $$ where $\varepsilon>0$ is a small noise parameter, $\{\xi_n\}$ is a sequence of independent standard Gaussian random variables, and $\varphi_{f,g}$ is constructed from two continuous nondecreasing maps $f,g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfying $f(0)=g(0)=0.$ We consider two classes of such maps. In the first class, the expansive behavior persists up to the endpoints of the interval. In the second class, the maps are expansive on inner intervals and linear and contractive on the corresponding outer intervals. For the first class, we prove that the logarithmic growth rate of the expected exit time vanishes as $\varepsilon\to 0.$ For the second class, we determine the exact logarithmic rate of the finite-horizon exit probability and derive an explicit upper bound for the expected exit time. More precisely, our analysis shows that the inner expansive nonlinear part of the drift has zero variational cost, while in the contractive case the outer linear part determines the explicit exponential rate. This extends earlier estimates obtained for particular piecewise-linear autoregressive models to broader classes of monotone nonlinear processes.