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Rough limit results for level-crossing probabilities

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Harri Nyrhinen
Harri Nyrhinen*
Affiliation:
Pohjola Insurance Company Ltd.
*
∗ Postal address: Pohjola Insurance Company Ltd., Lapinmäentie 1, 00013 Pohjola, Finland.
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Abstract

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

Keywords

LARGE DEVIATIONS THEORYINSURANCE MATHEMATICS

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 373 - 382
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

The main part of this paper was written during a research project in the Rolf Nevanlinna Institute with the support of the Foundation for the Promotion of the Actuarial Profession.

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