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Large deviations for the time of ruin

Part of:Limit theoremsStochastic processes

Published online by Cambridge University Press:  14 July 2016

Harri Nyrhinen
Harri Nyrhinen*
Affiliation:
University of Helsinki
*
∗ Postal address: Department of Mathematics, PO Box 4, FIN 00014, University of Helsinki, Finland. Email address: nyrhinen@helsinki.fi.
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Abstract

Let {Y n | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n > M} (T = +∞ if Y n ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {P M | M > 0} by P M (B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Y n }, the family {P M } satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y ⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.

Keywords

Ruin problemlarge deviations principleimportance sampling

MSC classification

Primary:60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:60F10: Large deviations

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999, pp. 733 - 746
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Supported by the Research Grants Committee of the University of Helsinki.

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