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 › peer-review

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.

Translated title of the contribution	Optimale Rückversicherung für Gerber Shiu Funktionen im Cramér-Lundberg Model
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

https://arxiv.org/pdf/1809.00990.pdf

Cite this

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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",

publisher = "Elsevier B.V.",

number = "87",

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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

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

Other files and links

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

PDMP based risk models

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

Cite this

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

Abstract

Keywords

ASJC Scopus subject areas

Fields of Expertise

Access to Document

Other files and links

Projects

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

Activities

PDMP based risk models

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

Cite this