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Refraction Strategies

  • Andreas E. Kyprianou
Chapter
Part of the EAA Series book series (EAAS)

Abstract

We consider the refracted Cramér–Lundberg process, that is, the solution to the stochastic differential equation (SDE)
$$\mathrm {d}Z_t = \mathrm {d}X_t - \delta \mathbf{1}_{(Z_t >\mathrm{b})}\,\mathrm {d}t, \quad t\geq 0, $$
for some threshold b≥0. We charge ourselves with the task of providing identities for the probability of ruin as well as the expected present value of dividends paid until ruin. It turns out that all identities can be written in terms of the scale functions of two different processes. One scale function comes from the Cramér–Lundberg process X and the other from the same Cramér–Lundberg process but with premium rate reduced by δ.

Keywords

Stochastic Differential Equation Premium Rate Strong Markov Property Natural Filtration Strong Markov Process 
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.

References

  1. Gerber, H.U., Shiu, E.S.W.: On optimal dividends: From reflection to refraction. J. Comput. Appl. Math. 186, 4–22 (2006a) MathSciNetCrossRefzbMATHGoogle Scholar
  2. Gerber, H.U., Shiu, E.S.W.: On optimal dividend strategies in the compound Poisson model. North Am. Actuar. J. 10, 76–93 (2006b) MathSciNetCrossRefGoogle Scholar
  3. Kyprianou, A.E., Loeffen, R.L.: Refracted Lévy processes. Ann. Inst. H. Poincaré. 46, 24–44 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. Kyprianou, A.E., Loeffen, R.L., Pérez, J.L.: Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Probab. 49, 150–166 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  5. Kyprianou, A.E., Pardo, J.C., Pérez, J.L.: Occupation times of refracted Lévy processes. Journal of Theoretical Probability (2013). doi: 10.1007/s10959-013-0501-4 Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Andreas E. Kyprianou
    • 1
  1. 1.Department of Mathematical SciencesUniversity of BathBathUnited Kingdom

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