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A dual risk model with additive and proportional gains: ruin probability and dividends

Part of: Stochastic systems and control Stochastic processes Markov processes

Published online by Cambridge University Press:  08 February 2023

Onno Boxma ,
Esther Frostig  and
Zbigniew Palmowski Open the ORCID record for Zbigniew Palmowski [Opens in a new window]
Onno Boxma*
Affiliation:
Eindhoven University of Technology
Esther Frostig*
Affiliation:
University of Haifa
Zbigniew Palmowski*
Affiliation:
Wrocław University of Science and Technology
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands. Email address: o.j.boxma@tue.nl
**Postal address: Department of Statistics, Haifa University, Haifa, Israel. Email address: frostig@stat.haifa.ac.il
***Postal address: Department of Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland. Email address: zbigniew.palmowski@pwr.edu.pl
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Abstract

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

Keywords

Dual risk modelruin probabilitytime to ruindividend

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 549 - 580
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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