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Optimal dividends with an affine penalty

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Abstract

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.

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Notes

  1. If we are referring to an insurance company, then “expenses” refers to aggregate claims.

  2. Similarly, \(\tau ^{d, L} = \inf \{t \ge 0: X_t^{d, L} < 0 \} = \inf \{t \ge 0: X_t^{d, L} \le 0 \}\) is the time of ruin of the dual risk process \(X^{d, L}\). We can replace < by \(\le \) in the definition of \(\tau ^{d, L}\) because of the continual negative drift \(-\, c\) in the dual risk process.

  3. We write negativek and positiveq in the penalty function because when ruin occurs, the deficit x will be negative. Also, we write P(x) with these signs so that increasing k and q leads to “more penalty” in an intuitive sense. Otherwise, if we were to write \(+\, k\) in place of \(-\, k\), increasing k would increase the reward for ruin; instead, we want to think of P(x) as a penalty function. Alternatively, we could have written \(P(x) = k - qx\) and subtracted P(x) in (2.4).

  4. Loeffen and Renaud [25] slightly erred when determining the condition for which paying all surplus is optimal in the compound Poisson case. Essentially they forgot to include the second case in Theorem 3.2.

  5. When Y is distributed according to the exponential with parameter \(\alpha \), then this positivity condition becomes \(\lambda - \alpha c > 0\).

  6. Note that \(k = 0\) is consistent with (4.11) because we assume that \(\lambda - \alpha c > 0\).

  7. In this section, we use a subscript or superscript d to distinguish quantities in the dual risk model from the corresponding ones in the classical risk model. So, for example, we will write \(R_1^d\) to mean \(R_1\), as given in (3.8), with c replaced by \(c_d\).

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Acknowledgements

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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  1. School of Mathematical Sciences, Nanjing Normal University, Jiangsu, 210023, China

    Zhibin Liang

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, United States

    Virginia R. Young

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Correspondence to Zhibin Liang.

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Zhibin Liang thanks the National Natural Science Foundation of China (Grant No. 11471165) for financial support.

Virginia R. Young thanks the Nesbitt Chair of Actuarial Mathematics for partial financial support.

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Liang, Z., Young, V.R. Optimal dividends with an affine penalty. J. Appl. Math. Comput. 60, 703–730 (2019). https://doi.org/10.1007/s12190-018-01233-y

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