{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T15:33:52Z","timestamp":1768404832707,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T00:00:00Z","timestamp":1690416000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Liaoning Province Department of Education","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}]},{"name":"Liaoning Province Department of Education","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}]},{"name":"Liaoning Province Department of Education","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}]},{"name":"Liaoning Province Department of Education","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}]},{"name":"Liaoning Province Department of Education","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}]},{"name":"Liaoning Province Department of Education","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}]},{"name":"List of Key Science and Technology Projects in Transportation Industry of Ministry of Transport","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}]},{"name":"Dalian Maritime University","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}]},{"name":"Dalian Maritime University","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}]},{"name":"Dalian Maritime University","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}]},{"name":"Dalian Maritime University","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}]},{"name":"Dalian Maritime University","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}]},{"name":"Dalian Maritime University","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}]},{"name":"Chinese Ministry of Education","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}]},{"name":"Chinese Ministry of Education","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}]},{"name":"Chinese Ministry of Education","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}]},{"name":"Chinese Ministry of Education","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}]},{"name":"Chinese Ministry of Education","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}]},{"name":"Chinese Ministry of Education","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["LJKZ0066"],"award-info":[{"award-number":["LJKZ0066"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["2021-BS-076"],"award-info":[{"award-number":["2021-BS-076"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["2022-MS3-102"],"award-info":[{"award-number":["2022-MS3-102"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["3132023721"],"award-info":[{"award-number":["3132023721"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["22YJC910011"],"award-info":[{"award-number":["22YJC910011"]}]},{"name":"China Postdoctoral Science Foundation Funded Project","award":["2023M733444"],"award-info":[{"award-number":["2023M733444"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper studies insurance companies\u2019 optimal reinsurance\u2013investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer\u2019s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox\u2013Ingersoll\u2013Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance\u2013investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance\u2013investment strategy is illustrated using numerical examples.<\/jats:p>","DOI":"10.3390\/axioms12080736","type":"journal-article","created":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T02:10:45Z","timestamp":1690510245000},"page":"736","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Optimal Reinsurance\u2013Investment Strategy Based on Stochastic Volatility and the Stochastic Interest Rate Model"],"prefix":"10.3390","volume":"12","author":[{"given":"Honghan","family":"Bei","sequence":"first","affiliation":[{"name":"School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China"},{"name":"Collaborative Innovation Centre for Transport Study, Dalian Maritime University, Dalian 116026, China"},{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]},{"given":"Qian","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China"}]},{"given":"Yajie","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China"}]},{"given":"Wenyang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China"},{"name":"Collaborative Innovation Centre for Transport Study, Dalian Maritime University, Dalian 116026, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0674-0363","authenticated-orcid":false,"given":"Roberto","family":"Murcio","sequence":"additional","affiliation":[{"name":"Department of Geography, Birkbeck, London University, Malet Street, Bloomsbury, London WC1E 7HX, UK"},{"name":"Centre for Advanced Spatial Analysis, University College London, 90 Tottenham Court Road, London W1T 4TJ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"884","DOI":"10.1006\/jmaa.1996.0474","article-title":"Another Approach to the Existence of Value Functions of Stochastic Differential Games","volume":"204","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1137\/S0363012900378863","article-title":"Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions","volume":"41","author":"Soner","year":"2002","journal-title":"SIAM J. Control Optim."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1137\/S0363012900371083","article-title":"Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation","volume":"40","author":"Rami","year":"2002","journal-title":"SIAM J. Control Optim. Soc. Ind. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhu, J., and Li, S. (2020). Time-Consistent Investment and Reinsurance Strategies for Mean-Variance Insurers under Stochastic Interest Rate and Stochastic Volatility. Mathematics, 8.","DOI":"10.3390\/math8122183"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ali, I., and Khan, S.U. (2023). A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method. Mathematics, 11.","DOI":"10.3390\/math11061328"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1287\/moor.20.4.937","article-title":"Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin","volume":"20","author":"Browne","year":"1995","journal-title":"Math. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1016\/j.insmatheco.2005.06.009","article-title":"Optimal investment for insurer with jump-diffusion risk process","volume":"37","author":"Yang","year":"2005","journal-title":"Insur. Math. Econ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0167-6687(00)00049-4","article-title":"Optimal investment for insurers","volume":"27","author":"Hipp","year":"2000","journal-title":"Insur. Math. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1080\/10920277.2005.10596214","article-title":"Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift","volume":"9","author":"David","year":"2005","journal-title":"N. Am. Actuar. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s00186-007-0195-4","article-title":"Dynamic mean-variance problem with constrained risk control for the insurers","volume":"68","author":"Bai","year":"2008","journal-title":"Math. Methods Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ramadan, A.T., Tolba, A.H., and El-Desouky, B.S. (2022). A Unit Half-Logistic Geometric Distribution and Its Application in Insurance. Axioms, 11.","DOI":"10.3390\/axioms11120676"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Georgescu, I., and Kinnunen, J. (2020). Optimal Saving by Expected Utility Operators. Axioms, 9.","DOI":"10.3390\/axioms9010017"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/j.insmatheco.2012.09.009","article-title":"Optimal investment strategies for the HARA utility under the constant elasticity of variance model","volume":"51","author":"Jung","year":"2012","journal-title":"Insur. Math. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.insmatheco.2016.10.014","article-title":"Optimal consumption\u2013investment strategy under the Vasicek model: HARA utility and Legendre transform","volume":"72","author":"Chang","year":"2017","journal-title":"Insur. Math. Econ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.3934\/jimo.2019011","article-title":"Optimal Reinsurance-Investment Problem with Dependent Risks Based on Legendre Transform","volume":"16","author":"Zhang","year":"2020","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5767","DOI":"10.3934\/jimo.2022194","article-title":"Optimal Investment Strategies for Asset-Liability Management with Affine Diffusion Factor Processes and Hara Preferences","volume":"19","author":"Zhang","year":"2023","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_17","unstructured":"Cox, J.C. (1975). Notes on Option Pricing I: Constant Elasticity of Variance Diffusion, Unpublished Note, Standford University, Graduate School of Business."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1093\/rfs\/4.4.727","article-title":"Stock Price Distributions with Stochastic Volatility: An Analytic Approach","volume":"4","author":"Stein","year":"1991","journal-title":"Rev. Financ. Stud."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/rfs\/6.2.327","article-title":"A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options","volume":"6","author":"Heston","year":"1993","journal-title":"Rev. Financ. Stud."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1016\/j.insmatheco.2012.09.003","article-title":"Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model","volume":"51","author":"Gu","year":"2012","journal-title":"Insur. Math. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1016\/j.jmaa.2019.02.016","article-title":"Optimal time-consistent reinsurance-investment strategy with delay for an insurer under a defaultable market","volume":"474","author":"Wang","year":"2019","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s00186-017-0570-8","article-title":"Robust optimal investment and reinsurance problem for a general insurance company under Heston model","volume":"85","author":"Huang","year":"2017","journal-title":"Math. Methods Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.frl.2018.10.009","article-title":"Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model","volume":"30","author":"Zhu","year":"2019","journal-title":"Financ. Res. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"113082","DOI":"10.1016\/j.cam.2020.113082","article-title":"Optimal excess-of-loss reinsurance and investment problem with thinning dependent risks under Heston model","volume":"382","author":"Zhang","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.insmatheco.2019.11.003","article-title":"Open-loop equilibrium reinsurance-investment strategy under mean\u2013variance criterion with stochastic volatility","volume":"90","author":"Yan","year":"2020","journal-title":"Insur. Math. Econ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1967872","DOI":"10.1155\/2016\/1967872","article-title":"Explicit Solution of Reinsurance-Investment Problem for an Insurer with Dynamic Income under Vasicek Model","volume":"2016","author":"Sheng","year":"2016","journal-title":"Adv. Math. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5176172","DOI":"10.1155\/2019\/5176172","article-title":"Optimal Investment-Reinsurance Policy with Stochastic Interest and Inflation Rates","volume":"2019","author":"Zhang","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2789","DOI":"10.1080\/02331934.2021.1887179","article-title":"Mean-variance problem for an insurer with dependent risks and stochastic interest rate in a jump-diffusion market","volume":"71","author":"Yuan","year":"2022","journal-title":"Optimization"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1007\/s11009-018-9630-7","article-title":"Optimal Reinsurance-Investment Strategy Under Risks of Interest Rate, Exchange Rate and Inflation","volume":"20","author":"Guo","year":"2018","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s00186-017-0628-7","article-title":"Optimal mean\u2013variance investment and reinsurance problem for an insurer with stochastic volatility","volume":"88","author":"Sun","year":"2018","journal-title":"Math. Methods Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.insmatheco.2019.01.002","article-title":"Time-consistent investment-proportional reinsurance strategy with random coefficients for mean\u2013variance insurers","volume":"85","author":"Wang","year":"2019","journal-title":"Insur. Math. Econ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.insmatheco.2019.09.004","article-title":"Robust optimal reinsurance and investment strategies for an AAI with multiple risks","volume":"89","author":"Guan","year":"2019","journal-title":"Insur. Math. Econ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"71","DOI":"10.3934\/jimo.2018141","article-title":"Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework","volume":"16","author":"Zhang","year":"2020","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Grandell, J. (1991). Aspects of Risk Theory, Springer.","DOI":"10.1007\/978-1-4613-9058-9"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Yong, J., and Zhou, X.Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer.","DOI":"10.1007\/978-1-4612-1466-3"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0167-6687(03)00153-7","article-title":"Optimal investment strategies in the presence of a minimum guarantee","volume":"33","author":"Deelstra","year":"2003","journal-title":"Insur. Math. Econ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.insmatheco.2014.05.004","article-title":"Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework","volume":"57","author":"Guan","year":"2014","journal-title":"Insur. Math. Econ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/j.insmatheco.2008.03.004","article-title":"Stochastic optimal control of DC pension funds","volume":"42","author":"Gao","year":"2008","journal-title":"Insur. Math. Econ."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/736\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:20:55Z","timestamp":1760127655000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/736"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,27]]},"references-count":38,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["axioms12080736"],"URL":"https:\/\/doi.org\/10.3390\/axioms12080736","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,27]]}}}