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Article: A class of nonzero-sum investment and reinsurance games subject to systematic risks

TitleA class of nonzero-sum investment and reinsurance games subject to systematic risks
Authors
Issue Date2017
PublisherTaylor & Francis Scandinavia. The Journal's web site is located at http://www.tandf.co.uk/journals/titles/03461238.asp
Citation
Scandinavian Actuarial Journal, 2017, v. 2017 n. 8, p. 670-707 How to Cite?
AbstractRecently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint. Scandinavian Actuarial Journal 2014(7), 583–601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims; and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 327–343]. The main objective of each insurer is to maximize the expected utility of his terminal surplus relative to that of his competitor. Dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear integral-typed equations. Despite the complex structure, we establish the unique existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurers in a representative example of the constant absolute risk aversion insurers under a mild, time-independent condition. Furthermore, Nash equilibrium strategies and value functions admit closed forms. Numerical studies are also provided to illustrate the impact of the systematic risks on the Nash equilibrium strategies. Finally, we connect our results to that under the diffusion-approximated model by proving explicitly that the Nash equilibrium under the diffusion-approximated model is an -Nash equilibrium under the general Poisson risk model, thereby establishing that the analogous Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam, S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential investment and reinsurance games. Automatica 50(8), 2025–2037] serves as an interesting complementary case of the present framework.
Persistent Identifierhttp://hdl.handle.net/10722/245292
ISSN
2017 Impact Factor: 1.55
2015 SCImago Journal Rankings: 0.956
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorSiu, CC-
dc.contributor.authorYam, SCP-
dc.contributor.authorYang, H-
dc.contributor.authorZhao, H-
dc.date.accessioned2017-09-18T02:08:03Z-
dc.date.available2017-09-18T02:08:03Z-
dc.date.issued2017-
dc.identifier.citationScandinavian Actuarial Journal, 2017, v. 2017 n. 8, p. 670-707-
dc.identifier.issn0346-1238-
dc.identifier.urihttp://hdl.handle.net/10722/245292-
dc.description.abstractRecently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint. Scandinavian Actuarial Journal 2014(7), 583–601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims; and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 327–343]. The main objective of each insurer is to maximize the expected utility of his terminal surplus relative to that of his competitor. Dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear integral-typed equations. Despite the complex structure, we establish the unique existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurers in a representative example of the constant absolute risk aversion insurers under a mild, time-independent condition. Furthermore, Nash equilibrium strategies and value functions admit closed forms. Numerical studies are also provided to illustrate the impact of the systematic risks on the Nash equilibrium strategies. Finally, we connect our results to that under the diffusion-approximated model by proving explicitly that the Nash equilibrium under the diffusion-approximated model is an -Nash equilibrium under the general Poisson risk model, thereby establishing that the analogous Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam, S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential investment and reinsurance games. Automatica 50(8), 2025–2037] serves as an interesting complementary case of the present framework.-
dc.languageeng-
dc.publisherTaylor & Francis Scandinavia. The Journal's web site is located at http://www.tandf.co.uk/journals/titles/03461238.asp-
dc.relation.ispartofScandinavian Actuarial Journal-
dc.rightsThis is an electronic version of an article published in Scandinavian Actuarial Journal, 2017, v. 2017 n. 8, p. 670-707. Scandinavian Actuarial Journal is available online at: https://www.tandfonline.com/doi/full/10.1080/03461238.2016.1228542-
dc.titleA class of nonzero-sum investment and reinsurance games subject to systematic risks-
dc.typeArticle-
dc.identifier.emailYang, H: hlyang@hku.hk-
dc.identifier.authorityYang, H=rp00826-
dc.description.naturepostprint-
dc.identifier.doi10.1080/03461238.2016.1228542-
dc.identifier.hkuros278172-
dc.identifier.volume2017-
dc.identifier.issue8-
dc.identifier.spage670-
dc.identifier.epage707-
dc.identifier.isiWOS:000409225000002-
dc.publisher.placeSweden-

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