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Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by \(\alpha\)-stable Lévy processes. (English) Zbl 1309.60059

This paper proves the convergence, after rescaling, of the first passage times of an integrated stable Ornstein-Uhlenbeck process starting at zero with zero velocity, towards the first passage times of the background driving stable Lévy process. This generalizes a well-known result in the Brownian case. Since the driving process is discontinuous and the integrated Ornstein-Uhlenbeck process is continuous, the uniform topology or the classical Skorokhod topology are not appropriate to this framework and the authors use the less classical M1-topology. The main result is Theorem 3.1 in the general context of real Lévy processes, which is a nice limit theorem in the M1-topology.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes

References:

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