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On distributions of first passage times and optimal stopping of AR(1) sequences. (English. Russian original) Zbl 1204.62143
Theory Probab. Appl. 53, No. 3, 419-429 (2009); translation from Teor. Veroyatn. Primen. 53, No. 3, 458-471 (2008).
The paper deals with the problem of boundedness of the first passage times \(\tau_a\) and \(\tau_{a,b}\) of autoregressive sequences \(\{X_n\}_{n=0}^\infty\), where \(X_n=\lambda X_{n-1}+\eta_n\), \(X_0=x\), \(x\) and \(\lambda\) are constants, and \(\{\eta_n\}\) is a sequence of i.i.d. random variables (innovations), for the boundary, i.e., \[ \tau_a=\inf\{n\geq 0: X_n>a\}\;\text{and}\;\tau_{a,b}=\inf\{n\geq 0:X_n>a\text{ or} X_n<b\}, \] respectively [see C. Stein, A note on cumulative sums. Ann. Math Stat. 17, 498–499 (1946; Zbl 0063.07178)]. A series of generalizations of the results obtained by the author and N. Kordzakhia, [Martingales and first passage times of AR(1) sequences. Stochastics 80, No. 2-3, 197–210 (2008; Zbl 1148.60061)] has been obtained.
Among others, for the case \(|\lambda|<1\) a formula for \({\mathbf E}\tau_{a,b}\) has been given. A simple consequence of this is a lower bound and logarithmic asymptotics for \({\mathbf E}_x\tau_{a,-a}\) as \(a\rightarrow \infty\) for the case of Gaussian innovations \(\{\eta_n\}\) with zero mean [see F.K. Klebaner and R. Sh. Liptser, Large deviations for past-dependent recursions. Probl. Inf. Transm. 32, No. 4, 320-330 (1996); translation from Probl. Peredachi Inf. 32, No. 4, 23–34 (1996; Zbl 1037.60500), for an upper bound of \({\mathbf E}_x\tau_{a,-a}\), in this case]. Accuracy of the asymptotic approximation is illustrated by Monte Carlo simulations. An explicit formula is derived for the generating function of the first passage time for the case of AR(1)-sequences generated by innovations with exponential distributions. The latter formula is used to study an optimal stopping problem.

MSC:
62L15 Optimal stopping in statistics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G40 Stopping times; optimal stopping problems; gambling theory
60G42 Martingales with discrete parameter
65C05 Monte Carlo methods
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