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Article: On a class of stochastic models with two-sided jumps

TitleOn a class of stochastic models with two-sided jumps
Authors
KeywordsBusy period
Defective renewal equation
Dual risk model
Gerber-Shiu function
GI/G/1 queue
Idle period
Negative customers
Time of recovery
Time of ruin
Two-sided jumps
Issue Date2011
PublisherSpringer New York LLC. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0257-0130
Citation
Queueing Systems, 2011, v. 69 n. 1, p. 1-28 How to Cite?
AbstractIn this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber-Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48-72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber-Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber-Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue. © 2011 The Author(s).
Persistent Identifierhttp://hdl.handle.net/10722/145095
ISSN
2015 Impact Factor: 0.875
2015 SCImago Journal Rankings: 1.146
ISI Accession Number ID
Funding AgencyGrant Number
Faculty of Science
Department of Statistics and Actuarial Science
University Research Committee at the University of Hong Kong201103159001
Funding Information:

Part of the work was completed during the author's visit to the Department of Actuarial Science at the University of Lausanne, and the host's hospitality is greatly appreciated. The author would like to thank the anonymous referees and Professor Hansjorg Albrecher for helpful comments and suggestions as well as Professor Kristina Sendova for use of various references. Support for the author from a start-up fund provided by the Faculty of Science and the Department of Statistics and Actuarial Science as well as the Seed Funding for Basic Research (Project number: 201103159001) provided by the University Research Committee at the University of Hong Kong is also gratefully acknowledged.

References

 

DC FieldValueLanguage
dc.contributor.authorCheung, ECKen_HK
dc.date.accessioned2012-02-21T05:44:49Z-
dc.date.available2012-02-21T05:44:49Z-
dc.date.issued2011en_HK
dc.identifier.citationQueueing Systems, 2011, v. 69 n. 1, p. 1-28en_HK
dc.identifier.issn0257-0130en_HK
dc.identifier.urihttp://hdl.handle.net/10722/145095-
dc.description.abstractIn this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber-Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48-72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber-Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber-Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue. © 2011 The Author(s).en_HK
dc.languageengen_US
dc.publisherSpringer New York LLC. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0257-0130en_HK
dc.relation.ispartofQueueing Systemsen_HK
dc.rightsThe Author(s)en_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong Licenseen_US
dc.subjectBusy perioden_HK
dc.subjectDefective renewal equationen_HK
dc.subjectDual risk modelen_HK
dc.subjectGerber-Shiu functionen_HK
dc.subjectGI/G/1 queueen_HK
dc.subjectIdle perioden_HK
dc.subjectNegative customersen_HK
dc.subjectTime of recoveryen_HK
dc.subjectTime of ruinen_HK
dc.subjectTwo-sided jumpsen_HK
dc.titleOn a class of stochastic models with two-sided jumpsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4551/resserv?sid=springerlink&genre=article&atitle=On a class of stochastic models with two-sided jumps&title=Queueing Systems&issn=02570130&date=2011-09-01&volume=69&issue=1& spage=1&authors=Eric C. K. Cheungen_US
dc.identifier.emailCheung, ECK: eckc@hku.hken_HK
dc.identifier.authorityCheung, ECK=rp01423en_HK
dc.description.naturepublished_or_final_versionen_US
dc.identifier.doi10.1007/s11134-011-9228-zen_HK
dc.identifier.scopuseid_2-s2.0-79955806402en_HK
dc.identifier.hkuros186003-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-79955806402&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume69en_HK
dc.identifier.issue1en_HK
dc.identifier.spage1en_HK
dc.identifier.epage28en_HK
dc.identifier.eissn1572-9443en_US
dc.identifier.isiWOS:000298879400001-
dc.publisher.placeUnited Statesen_HK
dc.description.otherSpringer Open Choice, 21 Feb 2012en_US
dc.identifier.scopusauthoridCheung, ECK=24461272100en_HK
dc.identifier.citeulike9327001-

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