Skip to main content
SpringerLink
Log in

Dynamic mean-variance problem with constrained risk control for the insurers

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

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 (LQ) 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 (HJB) equations, which is different from that given in Zhou et al. (SIAM J Control Optim 35:243–253, 1997). Furthermore, by comparisons, we find that they are identical under the two risk models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bäuerle N (2005) Benchmark and mean-variance problems for insurers. Math Methods Oper Res 62: 159–165

    Article  MATH  MathSciNet  Google Scholar 

  • Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Boston

    MATH  Google Scholar 

  • Brémaud P (1981) Point processes and queues. Springer, New York

    MATH  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Choulli T, Taksar M, Zhou XY (2003) Optimal dividend distribution and risk control. SIAM J Control optim 41: 1946–1979

    Article  MATH  MathSciNet  Google Scholar 

  • Crandell MG, Lions P (1983) Viscosity solution of Hamilton–Jacobi equations. Trans Am Math Soc 277(1): 1–42

    Article  Google Scholar 

  • Fleming WH, Soner HM (1993) Controlled markov processes and viscosity solutions. Springer, Berlin

    MATH  Google Scholar 

  • Grandell J (1991) Aspects of risk theory. Springer, New York

    MATH  Google Scholar 

  • Hϕjgaard B, Taksar M (2004) Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quant Financ 4: 315–327

    Article  Google Scholar 

  • Li X, Zhou XY, Lim A (2002) Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM J Control Optim 40(5): 1540–1555

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Luenberger DG (1968) Optimization by vector space methods. Wiley, New York

    Google Scholar 

  • Markowitz H (1952) Portfolio selection. J Financ 7: 77–91

    Article  Google Scholar 

  • Merton RC (1972) An analytic derivation of the efficient frontier. J Financ Quant Anal 10: 1851–1872

    Article  Google Scholar 

  • Øksendal B, Sulem A (2005) Applied stochastic control of jump diffusion. Springer, Berlin

    Google Scholar 

  • 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

    MATH  Google Scholar 

  • 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

    MATH  Google Scholar 

  • Schmidli H (2001) Optimal proportional reinsurance policies in a dynamic setting. Scand Actuar J 1: 55–68

    Article  MathSciNet  Google Scholar 

  • Schmidli H (2002) On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab 12(3): 890–907

    Article  MATH  MathSciNet  Google Scholar 

  • Soner HM (1986a) Optimal control with state-space constrain.. I SAIM J Control Optim 24(3): 552–561

    Article  MATH  MathSciNet  Google Scholar 

  • Soner HM (1986b) Optimal control with state-space constrain.. I SAIM J Control Optim 24(6): 1110–1122

    Article  MATH  MathSciNet  Google Scholar 

  • 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

  • Wang N (2007) Optimal investment for an insurer with utility preference. Insur Math Econ 40: 77–84

    Article  MATH  Google Scholar 

  • Wang Z, Xia J, Zhang L (2007) Optimal investment for an insurer: the martingale approach. Insur Math Econ 40: 322–334

    Article  MATH  MathSciNet  Google Scholar 

  • Yang H, Zhang L (2005) Optimal investment for insurer with jump-diffusion risk process. Insur Math Econ 37: 615–634

    Article  MATH  Google Scholar 

  • Zhou XY, Li D (2000) Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl Math Optim 42: 19–33

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou XY, Yong JM, Li X (1997) Stochastic verification theorems within the framework of viscosity solutions. SIAM J Control Optim 35: 243–253

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China

    Lihua Bai

  2. Department of Finance, School of Economics, Nankai University, Tianjin, 300071, People’s Republic of China

    Huayue Zhang

Authors
  1. Lihua Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huayue Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lihua Bai.

Additional information

This work was supported by National Basic Research Program of China (973 Program) 2007CB814905 and National Natural Science Foundation of China (10571092).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bai, L., Zhang, H. Dynamic mean-variance problem with constrained risk control for the insurers. Math Meth Oper Res 68, 181–205 (2008). https://doi.org/10.1007/s00186-007-0195-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-007-0195-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation