Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus

Yasutaka Shimizu; Zhimin Zhang

doi:10.1016/j.insmatheco.2017.02.006

Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus

Yasutaka Shimizu, Zhimin Zhang^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

Abstract

Consider an insurance surplus process driven by a Lévy subordinator, which is observed at discrete time points. An estimator of the Gerber–Shiu function is proposed via the empirical Fourier transform of the Gerber–Shiu function. By evaluating its mean squared error, we show the L²-consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator.

Original language	English
Pages (from-to)	84-98
Number of pages	15
Journal	Insurance: Mathematics and Economics
Volume	74
DOIs	https://doi.org/10.1016/j.insmatheco.2017.02.006
Publication status	Published - 2017 May 1

Keywords

Estimation
Fourier inversion
Gerber–Shiu function
L-consistency
Lévy risk model

ASJC Scopus subject areas

Statistics and Probability
Economics and Econometrics
Statistics, Probability and Uncertainty

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10.1016/j.insmatheco.2017.02.006

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title = "Estimating Gerber–Shiu functions from discretely observed L{\'e}vy driven surplus",

abstract = "Consider an insurance surplus process driven by a L{\'e}vy subordinator, which is observed at discrete time points. An estimator of the Gerber–Shiu function is proposed via the empirical Fourier transform of the Gerber–Shiu function. By evaluating its mean squared error, we show the L2-consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator.",

keywords = "Estimation, Fourier inversion, Gerber–Shiu function, L-consistency, L{\'e}vy risk model",

author = "Yasutaka Shimizu and Zhimin Zhang",

note = "Funding Information: The two authors would like to thank two referees for their valuable suggestions which significantly improved the paper. Yasutaka Shimizu is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C), Grant Number JP15K05009. Zhimin Zhang acknowledges the support from the National Natural Science Foundation of China [11471058, 11661074], the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007] and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 16YJC910005). Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

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doi = "10.1016/j.insmatheco.2017.02.006",

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T1 - Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus

AU - Shimizu, Yasutaka

AU - Zhang, Zhimin

N1 - Funding Information: The two authors would like to thank two referees for their valuable suggestions which significantly improved the paper. Yasutaka Shimizu is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C), Grant Number JP15K05009. Zhimin Zhang acknowledges the support from the National Natural Science Foundation of China [11471058, 11661074], the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007] and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 16YJC910005). Publisher Copyright: © 2017 Elsevier B.V.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - Consider an insurance surplus process driven by a Lévy subordinator, which is observed at discrete time points. An estimator of the Gerber–Shiu function is proposed via the empirical Fourier transform of the Gerber–Shiu function. By evaluating its mean squared error, we show the L2-consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator.

AB - Consider an insurance surplus process driven by a Lévy subordinator, which is observed at discrete time points. An estimator of the Gerber–Shiu function is proposed via the empirical Fourier transform of the Gerber–Shiu function. By evaluating its mean squared error, we show the L2-consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator.

KW - Estimation

KW - Fourier inversion

KW - Gerber–Shiu function

KW - L-consistency

KW - Lévy risk model

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UR - http://www.scopus.com/inward/citedby.url?scp=85015987785&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2017.02.006

DO - 10.1016/j.insmatheco.2017.02.006

M3 - Article

AN - SCOPUS:85015987785

SN - 0167-6687

VL - 74

SP - 84

EP - 98

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

ER -

Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus

Abstract

Keywords

ASJC Scopus subject areas

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