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A class of non-zero-sum stochastic differential games between two mean–variance insurers under stochastic volatility

Published online by Cambridge University Press:  02 November 2022

Jiannan Zhang Open the ORCID record for Jiannan Zhang [Opens in a new window] ,
Ping Chen ,
Zhuo Jin  and
Shuanming Li
Jiannan Zhang
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Parkville, VIC 3010, Australia. E-mails: jiannanz2@student.unimelb.edu.au; pche@unimelb.edu.au; shli@unimelb.edu.au
Ping Chen
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Parkville, VIC 3010, Australia. E-mails: jiannanz2@student.unimelb.edu.au; pche@unimelb.edu.au; shli@unimelb.edu.au
Zhuo Jin
Affiliation:
Department of Actuarial Studies and Business Analytics, Macquarie University, North Ryde, NSW 2109, Australia. E-mail: zhuo.jin@mq.edu.au
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Parkville, VIC 3010, Australia. E-mails: jiannanz2@student.unimelb.edu.au; pche@unimelb.edu.au; shli@unimelb.edu.au
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Abstract

This paper studies the open-loop equilibrium strategies for a class of non-zero-sum reinsurance–investment stochastic differential games between two insurers with a state-dependent mean expectation in the incomplete market. Both insurers are able to purchase proportional reinsurance contracts and invest their wealth in a risk-free asset and a risky asset whose price is modeled by a general stochastic volatility model. The surplus processes of two insurers are driven by two standard Brownian motions. The objective for each insurer is to find the equilibrium investment and reinsurance strategies to balance the expected return and variance of relative terminal wealth. Incorporating the forward backward stochastic differential equations (FBSDEs), we derive the sufficient conditions and obtain the general solutions of equilibrium controls for two insurers. Furthermore, we apply our theoretical results to two special stochastic volatility models (Hull–White model and Heston model). Numerical examples are also provided to illustrate our results.

Keywords

Equilibrium strategyFBSDEsMean–varianceReinsuranceNon-zero-sum game
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Special Issue 2: Probability and stochastic modeling in actuarial science and related fields , April 2023 , pp. 491 - 517
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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