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Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

  • Xiaowen Zhou (a1)

Abstract

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := SX and Y := XI first exit level a, respectively; let τ(a) and κ(a) denote the times when X first reaches S τ(a) and I κ(a), respectively. The main results of this paper concern the distributions of (τ(a), S τ(a), τ(a), Ŷ τ(a)) and of (κ(a), I κ(a), κ(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

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Copyright

Corresponding author

Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada. Email address: zhou@alcor.concordia.ca

Footnotes

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Supported by an NSERC operating grant.

Footnotes

References

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Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

  • Xiaowen Zhou (a1)

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