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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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