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Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

Part of: Stochastic processes

Published online by Cambridge University Press:  15 July 2020

Irmina Czarna ,
Adam Kaszubowski ,
Shu Li  and
Zbigniew Palmowski Open the ORCID record for Zbigniew Palmowski [Opens in a new window]
Irmina Czarna*
Affiliation:
Wrocław University of Science and Technology
Adam Kaszubowski*
Affiliation:
University of Wrocław
Shu Li*
Affiliation:
Western University
Zbigniew Palmowski*
Affiliation:
Wrocław University of Science and Technology
*
*Postal address: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Centre, Wrocław University of Science and Technology, Poland
**Postal address: Mathematical Institute, University of Wrocław, Poland
***Postal address: Department of Statistical and Actuarial Sciences, Western University, Canada
*Postal address: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Centre, Wrocław University of Science and Technology, Poland
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Abstract

In this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

Keywords

Dividendsfluctuation theoryMarkov modulationOmega modelpotential measures

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G17: Sample path properties
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 404 - 432
Copyright
© Applied Probability Trust 2020

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