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1. Generalized scale functions for spectrally negative Lévy processes.

Author

Contreras, Jesús and Rivero, Víctor

Subjects
*LAPLACE transformation, *DIFFERENTIAL equations, *FUNCTIONALS, *MATHEMATICS, *BESSEL functions
Abstract

For a spectrally negative Lévy process, scale functions appear in the solution of twosided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, see e.g. Li and Palmowski (2018) and therefore the results in this work are an extension of them. [ABSTRACT FROM AUTHOR]

Published
2023
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2. Dividend barrier strategy: Proceed with caution.

Author

Sendova, Kristina P., Yang, Chen, and Zhang, Ruixi

Subjects
*LEVY processes, *PROBABILITY theory, *GENERALIZATION, *RANDOM walks, *MATHEMATICS
Abstract

We consider a Lévy risk process and a Sparre-Andersen risk process with Parisian ruin in the presence of a constant dividend barrier. We demonstrate that with few exceptions, ruin occurs with probability one. Subsequently, generalizations to certain dependent risk processes are discussed. Despite the mathematical nature of this paper, its goal is to convey some simple conclusions to the actuarial community. The reader may focus solely on the introduction and conclusion sections (Sections 1 and 5, respectively) as well as the numerical illustrations. [ABSTRACT FROM AUTHOR]

Published
2018
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3. Mathematics of the Bond Market: A Lévy Processes Approach: by Michał Barski and Jerzy Zabczyk, Cambridge University Press (2020). Hardback. ISBN 9781107101296.

Author

Grbac, Zorana and Horvath, Blanka

Subjects
*LEVY processes, *BOND market, *YIELD curve (Finance), *MATHEMATICS, *RANDOM walks, *LIBOR, *ARBITRAGE
Abstract

The Lévy-driven affine short rate models follow and a chapter on the completeness of the presented Lévy term structure models concludes. In conclusion, the book by M. Barski and J. Zabczyk on modelling fixed income markets based on Lévy processes is a textbook that fills perfectly a gap in the existing mathematical finance literature. The applications of Lévy processes in mathematical finance go back to the seminal paper of Merton ([4]), entitled "Option pricing when underlying stock returns are discontinuous", which galvanized the research landscape on Lévy models with respect to financial applications. [Extracted from the article]

Published
2021
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