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Dynamic mean-variance problem with constrained risk control for the insurers. (English) Zbl 1156.93037
Summary: In this paper, we study optimal reinsurance/new business and investment (no-shorting) strategy for the mean-variance problem in two risk models: a classical risk model and a diffusion model. The problem is firstly reduced to a stochastic linear-quadratic control problem with constraints. Then, the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton-Jacobi-Bellman equations, which is different from that given in X. Y. Zhou, J. Yong and X. Li [SIAM J. Control Optimization 35, No. 1, 243–253 (1997; Zbl 0880.93059)]. Furthermore, by comparisons, we find that they are identical under the two risk models.

MSC:
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
60J60 Diffusion processes
49L20 Dynamic programming in optimal control and differential games
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[1] Bäuerle N (2005) Benchmark and mean-variance problems for insurers. Math Methods Oper Res 62: 159–165 · Zbl 1101.93081 · doi:10.1007/s00186-005-0446-1
[2] Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Boston · Zbl 0890.49011
[3] Brémaud P (1981) Point processes and queues. Springer, New York
[4] 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 · Zbl 0846.90012 · doi:10.1287/moor.20.4.937
[5] Choulli T, Taksar M, Zhou XY (2003) Optimal dividend distribution and risk control. SIAM J Control optim 41: 1946–1979 · Zbl 1084.91047 · doi:10.1137/S0363012900382667
[6] Crandell MG, Lions P (1983) Viscosity solution of Hamilton–Jacobi equations. Trans Am Math Soc 277(1): 1–42 · doi:10.1090/S0002-9947-1983-0690039-8
[7] Fleming WH, Soner HM (1993) Controlled markov processes and viscosity solutions. Springer, Berlin · Zbl 0773.60070
[8] Grandell J (1991) Aspects of risk theory. Springer, New York · Zbl 0717.62100
[9] Hjgaard B, Taksar M (2004) Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quant Financ 4: 315–327 · doi:10.1088/1469-7688/4/3/007
[10] Li X, Zhou XY, Lim A (2002) Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM J Control Optim 40(5): 1540–1555 · Zbl 1027.91040 · doi:10.1137/S0363012900378504
[11] Lions P (1983) Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. II. Viscosity solutions and Uniqueness. Comm Partial Diff Equ 8(11): 1229–1276 · Zbl 0716.49023 · doi:10.1080/03605308308820301
[12] Luenberger DG (1968) Optimization by vector space methods. Wiley, New York
[13] Markowitz H (1952) Portfolio selection. J Financ 7: 77–91 · doi:10.2307/2975974
[14] Merton RC (1972) An analytic derivation of the efficient frontier. J Financ Quant Anal 10: 1851–1872 · doi:10.2307/2329621
[15] Øksendal B, Sulem A (2005) Applied stochastic control of jump diffusion. Springer, Berlin · Zbl 1074.93009
[16] Sayah A (1991a) Équations d’Hamilton–Jacobi du premier ordre avec termes intégro différetiels. I. Unicité des solutions de viscosité. Comm Partial Diff Equ 16(6–7): 1075–1093 · Zbl 0742.45005 · doi:10.1080/03605309108820790
[17] Sayah A (1991b) Équations d’Hamilton–Jacobi du premier ordre avec termes intégro différetiels. II. Existence des solutions de viscosité. Comm Partial Diff Equ 16(6–7): 1075–1093 · Zbl 0742.45005 · doi:10.1080/03605309108820790
[18] Schmidli H (2001) Optimal proportional reinsurance policies in a dynamic setting. Scand Actuar J 1: 55–68 · Zbl 0971.91039 · doi:10.1080/034612301750077338
[19] Schmidli H (2002) On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab 12(3): 890–907 · Zbl 1021.60061 · doi:10.1214/aoap/1031863173
[20] Soner HM (1986a) Optimal control with state-space constrain.. I SAIM J Control Optim 24(3): 552–561 · Zbl 0597.49023 · doi:10.1137/0324032
[21] Soner HM (1986b) Optimal control with state-space constrain.. I SAIM J Control Optim 24(6): 1110–1122 · Zbl 0619.49013 · doi:10.1137/0324067
[22] Soner HM (1988) Optimal control of jump-markov process and viscosity solutions. Stochastic differential system, stochastic control and applications (Minneapolis, Minn., 1986), IMA A volumes in mathematics and its applications, vol 10. Springer, New York, pp 501–511
[23] Wang N (2007) Optimal investment for an insurer with utility preference. Insur Math Econ 40: 77–84 · Zbl 1273.91431 · doi:10.1016/j.insmatheco.2006.02.008
[24] Wang Z, Xia J, Zhang L (2007) Optimal investment for an insurer: the martingale approach. Insur Math Econ 40: 322–334 · Zbl 1141.91470 · doi:10.1016/j.insmatheco.2006.05.003
[25] Yang H, Zhang L (2005) Optimal investment for insurer with jump-diffusion risk process. Insur Math Econ 37: 615–634 · Zbl 1129.91020 · doi:10.1016/j.insmatheco.2005.06.009
[26] Zhou XY, Li D (2000) Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl Math Optim 42: 19–33 · Zbl 0998.91023 · doi:10.1007/s002450010003
[27] Zhou XY, Yong JM, Li X (1997) Stochastic verification theorems within the framework of viscosity solutions. SIAM J Control Optim 35: 243–253 · Zbl 0880.93059 · doi:10.1137/S0363012995279973
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