Optimal dividends with an affine penalty

  • Zhibin LiangEmail author
  • Virginia R. Young
Original Research


We find the optimal dividend strategy in two related risk models under an affine penalty for ruin. The first risk model is the classical Cramér–Lundberg risk model, and the second is the so-called dual risk model. Under both models, for exponentially distributed jumps, we show that the optimal dividend strategy is a barrier strategy and obtain the barrier explicitly. Moreover, we prove that the optimal barrier increases with respect to the parameters of the affine penalty, while the penalized value function decreases with respect to the penalty. We also compute the expected time until ruin and show that the expected time of ruin increases with respect to the parameters of the affine penalty. Finally, we present some numerical examples to demonstrate the relationship between the results for the classical and dual risk models. Our main contributions are in comparing the classical and dual risk models side-by-side and in obtaining explicit expressions for the penalized value functions, the optimal barriers, and the expected times of ruin. Also, we contrast the free-boundary condition associated with the barrier strategies in the two models.


Optimal dividend strategy Ruin penalty Classical risk model Dual risk model Instantaneous control Impulse control 

Mathematics Subject Classification

93E20 91B30 

JEL Classification

C58 C61 G22 



The authors would like to thank the anonymous referees for their careful reading and helpful comments on an earlier version of this paper, which led to a considerable improvement of the presentation of the work.


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Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesNanjing Normal UniversityJiangsuChina
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUnited States

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