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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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