Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

Michael Julius Preischl; Stefan Thonhauser

doi:https://doi.org/10.1016/j.insmatheco.2019.04.002

Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

Michael Julius Preischl, Stefan Thonhauser

Research output: Contribution to journal › Article

Abstract

Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions)in a Cramér–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem's Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.

Original language	English
Pages (from-to)	82-91
Number of pages	10
Journal	Insurance / Mathematics & economics
Volume	87
Issue number	87
Early online date	Apr 2019
DOIs	https://doi.org/10.1016/j.insmatheco.2019.04.002
Publication status	Published - 2019

Keywords

Cramér-Lundberg model
Dynamic reinsurance
Gerber–Shiu functions
Optimal stochastic control
Policy iteration

ASJC Scopus subject areas

Economics and Econometrics
Statistics and Probability
Statistics, Probability and Uncertainty

Fields of Expertise

Information, Communication & Computing

Access to Document

https://doi.org/10.1016/j.insmatheco.2019.04.002

Fingerprint Dive into the research topics of 'Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model'. Together they form a unique fingerprint.

1 Active

Special Research Area (SFB) F55 Quasi-Monte Carlo Methods: Theory and Applications
Grabner, P., Tichy, R., Kusner, W. B., Ziefle, J., Brauchart, J., Iaco, M. R. & Aistleitner, C.
1/02/14 → 31/01/22
Project: Research project

1 Talk at conference or symposium

Optimal Reinsurance for Gerber-Shiu Functions in the Cramer-Lundberg Model
Stefan Michael Thonhauser (Speaker)
11 Jul 2019
Activity: Talk or presentation › Talk at conference or symposium › Science to science

Cite this

@article{c5f10c63d7784f47a3aad3361fbd6be9,

title = "Optimal reinsurance for Gerber–Shiu functions in the Cram{\'e}r–Lundberg model",

abstract = "Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions)in a Cram{\'e}r–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem's Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.",

keywords = "Cram{\'e}r-Lundberg model, Dynamic reinsurance, Gerber–Shiu functions, Optimal stochastic control, Policy iteration",

author = "Preischl, {Michael Julius} and Stefan Thonhauser",

year = "2019",

doi = "https://doi.org/10.1016/j.insmatheco.2019.04.002",

language = "English",

volume = "87",

pages = "82--91",

journal = "Insurance / Mathematics & economics",

issn = "0167-6687",

publisher = "Elsevier B.V.",

number = "87",

}

TY - JOUR

T1 - Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

AU - Preischl, Michael Julius

AU - Thonhauser, Stefan

PY - 2019

Y1 - 2019

N2 - Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions)in a Cramér–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem's Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.

AB - Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions)in a Cramér–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem's Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.

KW - Cramér-Lundberg model

KW - Dynamic reinsurance

KW - Gerber–Shiu functions

KW - Optimal stochastic control

KW - Policy iteration

UR - http://www.scopus.com/inward/record.url?scp=85064859764&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.insmatheco.2019.04.002

DO - https://doi.org/10.1016/j.insmatheco.2019.04.002

M3 - Article

VL - 87

SP - 82

EP - 91

JO - Insurance / Mathematics & economics

JF - Insurance / Mathematics & economics

SN - 0167-6687

IS - 87

ER -

Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

Abstract

Keywords

ASJC Scopus subject areas

Fields of Expertise

Access to Document

Special Research Area (SFB) F55 Quasi-Monte Carlo Methods: Theory and Applications

Optimal Reinsurance for Gerber-Shiu Functions in the Cramer-Lundberg Model

Cite this