Equilibrium reinsurance-investment strategy for mean-variance insurers under state dependent risk aversion

• Autorzy: Alia Ishak , Chighoub Farid
• Języki publikacji: EN

Abstrakty

EN

In this work, we study the equilibrium reinsurance/ new business and investment strategy for mean-variance insurers, under the assumption that the risk aversion is a function of current wealth level. The surplus of the agents is represented by a sum of a compound process and a linear premium perturbed with a Brownian component. The financial market consists of one riskless asset and a multiple risky assets whose price processes are driven by Poisson random measures and independent Brownian motions. We characterize explicit expressions for the time-consistent Nash equilibrium strategy and the equilibrium value function via a forward-backward stochastic system and an equilibrium condition. An interesting feature of these FBSDEs is that a time parameter is involved, so that they form a flow of FBSDEs. Furthermore, a feedback representation of an equilibrium solution is derived. This solution provides a tool for comparing the equilibrium strategy with those derived in other papers, where some special cases were studied by the dynamic programming argument.

Słowa kluczowe

EN

time inconsistency; mean-variance criterion; investment-reinsurance strategy; insurer; equilibrium strategy; forward-backward stochastic differential equation

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 4
• Strony: 489--523
• Opis fizyczny: Bibliogr. 45 poz., rys.

Twórcy

autor

Alia Ishak

• Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145 Biskra (07000), Algeria

autor

Chighoub Farid

• Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145 Biskra (07000), Algeria

Bibliografia

• Alia, I., Chighoub, F. and Sohail, A. (2016) A Characterization of Equilibrium Strategies in Continuous-Time Mean-Variance Problems for Insurers. Insurance: Mathematics and Economics, 68, 212–223.
• Bai, L.H. and Zhang, H.Y. (2008) Dynamic mean-variance problem with constrained risk control for the insurers. Mathematical Methods of Operations Research, 68, 181–205.
• Basak, S. and Chabakauri, G. (2010) Dynamic mean-variance asset allocation. Review of Financial Studies, 23, 2970-3016.
• Bäuerle, N. (2005) Benchmark and mean-variance problems for insurers. Mathematical Methods of Operations Research, 62, 159–165.
• Björk, T. and Murgoci, A. (2008) A general theory of Markovian time-inconsistent stochastic control problems, SSRN:1694759.
• Björk, T., Murgoci, A. and Zhou, X. (2012) Mean-variance portfolio optimization with state dependent risk aversion. Mathematical Finances. http://dx.doi.org/10.1111/j.1467-9965.2011.00515.x
• Björk, T., Murgoci, A. and Zhou, X.Y (2014) Mean-variance portfolio optimization with state-dependent risk aversion. Mathematical Finance, 24(1), 1-24.
• Browne, S.(1995) Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin. Math. Oper. Res., 20(4), 937–958.
• Cao Y. and Wan, N. (2009) Optimal proportional reinsurance and investment based on Hamilton–Jacobi–Bellman equation. Insurance: Mathematics and Economics, 45, 157–162.
• Chighoub, F. and Mezerdi, B. (2013) A stochastic maximum principle in mean-field optimal control problems for jump diffusions. Arab J. Math. Sci., 19(2), 223–241.
• Czichowsky, C. (2012) Time-consistent mean-variance porftolio selection in discrete and continuous time, arXiv:1205.4748v1.
• Delong, L. and Gerrard, R. (2007) Mean-variance portfolio selection for a nonlife insurance company. Mathematical Methods of Operations Research, 66, 339–367.
• Ekeland, I. and Lazrak, A. (2008)Equilibrium policies when preferences are time-inconsistent, arXiv:0808.3790v1.
• Ekeland, I. and Pirvu, T.A. (2008) Investment and consumption without commitment. Mathematics and Financial Economics, 2, 57-86.
• Ekeland, I., Mbodji, O. and Pirvu, T.A. (2012) Time-consistent portfolio management. SIAM J. Financ. Math., 3, 1-32.
• Hipp, C., and Plum, M. (2000) Optimal investment for insurers. Insurance: Mathematics and Economics, 26, 215-228.
• Hu, Y., Jin, H. and Zhou, X.Y (2012) Time-inconsistent stochastic linear quadratic control. SIAM J. Control Optim., 50(3), 1548-1572.
• Hu, Y., Jin, H. and Zhou, X.Y (2017) The Uniqueness of Equilibrium for Time-Inconsistent Stochastic Linear–Quadratic Control. SIAM Journal on Control and Opti., 50(3), 1548-1572.
• Goldman, S.M. (1980) Consistent plans. Rev. Financial Stud., 47, 533-537.
• Grandell, J. (1991) Aspects of Risk Theory. Springer-Verlag, New York.
• Gu, M.D, Yang, Y.P., Li, S.D. and Zhang, J.Y. (2010) Constant elasticity of variance model for proportional reinsurance and investment strategies. Insurance: Mathematics and Economics, 46 (3), 580–587.
• Krusell, P. and Smith, A. (2003) Consumption and savings decisions with quasi-geometric discounting. Econometrica, 71, 366–375.
• Li, D. and Ng, W.L. (2000) Optimal dynamic portfolio selection: Multiperiod mean-variance formulation. Mathematical Finance, 10, 387-406.
• Li, D., Rong, X. and Zhao, H. (2015) Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model. Journal of Computational and Applied Mathematics, 283, 142-162.
• Li, Z.F., Zeng, Y. and Lai, Y.Z. (2012) Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance: Mathematics and Economics, 51, 191–203.
• Markowitz, H.M. (1952) Portfolio selection. Journal of Finance, 7, 77–91.
• Meng, Q (2014) General linear quadratic optimal stochastic control problem driven by a Brownian motion and a Poisson random martingale measure with random coefficients. Stochastic Analysis and Applications, 32(1), 88-109.
• Øksendal, B. and Sulem, A. (2007) Applied Stochastic Control of Jump Diffusions, Second Edition. Springer.
• Phelps, E.S. and Pollak, R.A. (1968) On second-best national saving and game-equilibrium growth. Review of Economic Studies, 35, 185-199.
• Pollak, R. (1968) Consistent planning. Rev. Financial Studies, 35, 185–199.
• Promislow, D.S. and Young, V.R. (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 9(3), 109–128.
• Strotz, R. (1955)Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23, 165–180.
• Tang, S. and Li, X. (1994) Necessary conditions for optimal control for stochastic systems with random jumps. SIAM J. Control Optim., 32,1447-1475.
• Wang, Z., Xia, J. and Zhang, L. (2007) Optimal investment for an insurer: the martingale approach. Insurance: Mathematics and Economics, 40, 322–334.
• Wang, N.(2007) Optimal investment for an insurer with utility preference. Insurance: Mathematics and Economics, 40, 77–84.
• Wu, H.L. (2013) Time-consistent strategies for a multiperiod mean-variance portfolio selection problem. J. Appl. Math., Article ID 841627, 13 pages.
• Xu, L., Wang, R.M. and Yao, D.J. (2008) On maximizing the expected terminal utility by investment and reinsurance. Journal of Industrial and Management Optimization, 4(4), 801–815.
• Yang, H. and Zhang, L. (2005) Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37, 615–634.
• Yong, J. and Zhou, X.Y. (1999) Stochastic Controls, Hamiltonian Systems and HJB Equations. Springer Verlage.
• Li, Y. and Li, Z. (2013) Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion. Insurance: Mathematics and Economics, 53(1), 86-97.
• Zeng, Y. and Li, Z.F. (2012) Optimal reinsurance–investment strategies for insurers under mean-CaR criteria. Journal of Industrial and Management Optimization, 8(3), 673–690.
• Zeng, Y., Li, Z.F. and Liu, J.J. (2011) Optimal strategies of benchmark and mean–variance portfolio selection problems for insurers. Journal of Industrial and Management Optimization, 6, 483–496.
• Zeng, Y. and Li, Z.F. (2011) Optimal time-consistent investment and reinsurance policies for mean–variance insurers. Insurance: Mathematics and Economics, 49, 145–154.
• Zeng, Y., Li, Z.F. and Lai, Y.Z. (2013) Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps. Insurance: Mathematics and Economics, 52(3), 498-507.
• Zhou, X.Y. and Li, D. (2000) Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework. Appl. Math. Optim., 42, 19-33.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).