Refraction Strategies

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


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.


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