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Martingales and first passage times of AR(1) sequences. (English) Zbl 1148.60061
The exponential boundedness for the first passage time $$\tau_a$$ over a level for an ergodic autoregressive sequence $X_n=\lambda X_{n-1}+\eta_n,\quad n\geq 1, \quad 0<\lambda<1,$ where $$\{\eta_n\}$$ are i.i.d. random variables, is shown. By using martingale arguments under some conditions the exact representation of $$E\tau_a$$ is obtained. This representation is used for obtaining the upper bound for $$E\tau_a$$ in the general case.

##### MSC:
 60J55 Local time and additive functionals 49J15 Existence theories for optimal control problems involving ordinary differential equations
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 [1] DOI: 10.1080/15326340500294702 · Zbl 1083.60064 · doi:10.1080/15326340500294702 [2] Borovkov A.A., Applications of Mathematics (1973) · Zbl 0283.60029 [3] Doney R.A., Fluctuation Theory for Levy Processes (2005) [4] Frisen M., Sequential Anal. 25 pp 379– (2006) [5] DOI: 10.2307/1426747 · Zbl 0443.60037 · doi:10.2307/1426747 [6] M. Jacobsen, Exit times for a class of autoregressive sequences and random walks, Preprint No 5, Department of Applied Mathematics and Statistics, University of Copenhagen, 2007 [7] M. Jacobsen and A. Jensen, Exit times for a class of piecewise exponential Markov processes with two-sided jumps, Preprint No 5, Department of Applied Mathematics and Statistics, University of Copenhagen, 2006 · Zbl 1125.60080 [8] Jeffreys H., Methods of Mathematical Physics pp 406–, 3. ed. (1988) [9] Kyprianou A.E., Introductory Lectures on Fluctuations of Lèvy Processes with Applications (2006) · Zbl 1104.60001 [10] DOI: 10.1137/1135035 · Zbl 0723.60044 · doi:10.1137/1135035 [11] Novikov A.A., Theory Probab. Appl. 48 pp 340– (2003) [12] Novikov A.A., Proc. Steklov Math. Inst. 202 pp 169– (1993) [13] DOI: 10.1016/j.probengmech.2004.04.005 · doi:10.1016/j.probengmech.2004.04.005 [14] DOI: 10.2307/2291599 · Zbl 0881.62105 · doi:10.2307/2291599 [15] Sukparungsee S., KMITL Sci. J.: Int. J. Sci. Appl. Sci. 6 pp 373– (2006) [16] DOI: 10.2307/1426858 · Zbl 0417.60073 · doi:10.2307/1426858
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