Blätter der DGVFM

, Volume 30, Issue 1, pp 1–13 | Cite as

Optimal control of capital injections by reinsurance in a diffusion approximation

  • Julia Eisenberg
  • Hanspeter SchmidliEmail author
Original Research Paper


In this paper we consider a diffusion approximation to a classical risk process, where the claims are reinsured by some reinsurance with deductible b ∈ [0,b̃], where b = b̃ means “no reinsurance” and b = 0 means “full reinsurance”. The cedent can choose an adapted reinsurance strategy (b t ) t ≥0, i. e. the deductible can be changed continuously. In addition, the cedent has to inject fresh capital in order to keep the surplus positive. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton–Jacobi–Bellman approach. Some examples illustrate the method.


Diffusion Approximation Surplus Process Optimal Dividend Capital Injection Proportional Reinsurance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asmussen S (2000) Ruin Probabilities. World Scientific, SingaporeGoogle Scholar
  2. 2.
    Azcue P, Muler N (2005) Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math Finance 15:261–308zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    de Finetti B (1957) Su un’impostazione alternativa della teoria collettiva del rischio. Trans XVth Int Congr Actuar 2:433–443Google Scholar
  4. 4.
    Gerber HU (1969) Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Schweiz Verein Versicherungsmath Mitt 69:185–228Google Scholar
  5. 5.
    Gerber HU (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monographs, PhiladelphiazbMATHGoogle Scholar
  6. 6.
    Grandell J (1991) Aspects of Risk Theory. Springer-Verlag, New YorkzbMATHGoogle Scholar
  7. 7.
    Hipp C (2003) Optimal dividend payment under a ruin constraint: Discrete time and state space. DGVFM 26:255–264CrossRefGoogle Scholar
  8. 8.
    Kulenko N, Schmidli H (2008) Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insur Math Econ 43:270–278CrossRefMathSciNetGoogle Scholar
  9. 9.
    Rogers LCG, Williams D (2000) Diffusions, Markov processes and martingales, vol 1. Cambridge University Press, CambridgeGoogle Scholar
  10. 10.
    Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic Processes for Insurance and Finance. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
  11. 11.
    Schmidli H (2001) Optimal proportional reinsurance policies in a dynamic setting. Scand Actuar J 55–68Google Scholar
  12. 12.
    Schmidli H (2008) Stochastic Control in Insurance. Springer-Verlag, LondonzbMATHGoogle Scholar
  13. 13.
    Shreve SE, Lehoczky JP, Gaver DP (1984) Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J Control Optim 22:55–75zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© DAV / DGVFM 2009

Authors and Affiliations

  1. 1.CologneGermany

Personalised recommendations