Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

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There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.

TidsskriftStochastic Processes and Their Applications
Sider (fra-til)105-123
StatusUdgivet - 2021

Bibliografisk note

Funding Information:
We are thankful to the two anonymous referees for their insightful comments. The second author gratefully acknowledges the financial support of Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk”.

Publisher Copyright:
© 2021 Elsevier B.V.

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