On the first entry time of a \(\mathbb{Z}_+\)-valued AR(1) process (Q1807968)

The aim of the paper is to study the following first exit time: \[ \tau(A)= \inf \{n\in \mathbb{N}\mid X_n\geq A\}, \qquad A\in \mathbb{N}, \] over a given level of a \(\mathbb{Z}_+\)-valued AR(1) process \((X_n\), \(n\geq 0)\). This class of stopping times arises naturally in the theory and applications of stochastic processes as well as in areas of statistics such as sequential analysis. The main aim of the paper is to derive an explicit formula for the expected value \(E(\tau(A))\) of the first exit time. Using the martingale theory and based on \textit{A. A. Novikov}'s approach [Theory Probab. Appl. 35, No. 2, 269-279 (1990); translation from Teor. Veroyatn. Primen. 35, No. 2, 282-292 (1990; Zbl 0702.60048)], the authors obtain a generalized Wald's equation that holds under a simple integrability condition. As an application, an asymptotic formula is given for \(E(\tau(A))\) of the AR(1) process with a thinned Poisson innovation.