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On point measures of [var epsilon]-upcrossings for stationary diffusions

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  • Grigelionis, Bronius

Abstract

We consider ergodic strictly stationary diffusion processes on an open interval (l,r)[subset, double equals]R1 with a predetermined stationary distribution H, named H-diffusions. Simple sufficient conditions for vague convergence of the time-normalized point measures of [var epsilon]-upcrossings of an H-diffusion to the Poisson measure are derived from the general result by Borkovec and Klüppelberg (Extremes 1(1) (1998) 47) in the terms of the stationary density and the scale function, describing an H-diffusion. Some examples are discussed in detail.

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  • Grigelionis, Bronius, 2003. "On point measures of [var epsilon]-upcrossings for stationary diffusions," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 403-410, February.
    • Handle: RePEc:eee:stapro:v:61:y:2003:i:4:p:403-410
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    References listed on IDEAS

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    1. Davis, Richard A., 1982. "Maximum and minimum of one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 13(1), pages 1-9, July.
    2. Tina Hviid Rydberg, 1999. "Generalized Hyperbolic Diffusion Processes with Applications in Finance," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 183-201, April.
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