Theory of Probability and Mathematical Statistics
DIVIDENDS WITH TAX AND CAPITAL INJECTION IN A SPECTRALLY NEGATIVE LEVY RISK MODEL
Abstract: We consider a risk model driven by a spectrally negative L'evy process. From the surplus dividends are paid and capital injections have to be made in order to keep the surplus positive. In addition, tax has to be paid for dividends, but injections lead to an exemption from tax. We generalize the results from [12, 13] and show that the optimal dividend strategy is a two-barrier strategy. The barrier depends on whether an immediate dividend would be taxed or not. For a risk process perturbed by diusion with exponentially distributed claim sizes, we show how the value function and the barriers can be determined.
Keywords: L'evy risk model, dividends, capital injections, tax, barrier strategy, Hamilton-Jacobi-Bellman equation, perturbed risk model.
1. S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scientic, Singapore, 2010.
2. H. Albrecher and J. Ivanovs, Linking dividends and capital injections | a probabilistic approach , Scand. Actuarial J. (2017), forthcoming.
3. F. Avram, Z. Palmowski, and M. R. Pistorius, On the optimal dividend problem for a spectrally negative L'evy process, Ann. Appl. Probab. 17 (2007), 156-180.
4. P. Azcue and N. Muler, Stochastic Optimization in Insurance, Springer, New York, 2014.
5. B. de Finetti, Su un' impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, vol. 2, 1957, 433-443.
6. H. U. Gerber, Entscheidungskriterien fur den zusammengesetzten Poisson-Prozess, Schweiz. Verein. Versicherungsmath. Mitt. 69 (1969), 185-228.
7. N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramer-Lundberg model with capital injections, Insurance Math. Econom. 43 (2008), 270-278.
8. Yu. Mishura and O. Ragulina, Ruin Probabilities: Smoothness, Bounds and Supermartingale Approach, ISTE Press Elsevier, London, 2016.
9. Yu. S. Mishura, O. Yu. Ragulina, and O. M. Stroev, Analytic property of innite-horizon survival probability in a risk model with additional funds, Theory Probab. Math. Statist. 91 (2015), 131-143.
10. T. Rolski, H. Schmidli, V. Schmidt, and J. L. Teugels, Stochastic Processes for Insurance and Finance, Wiley, Chichester, 1999.
11. H. Schmidli, Stochastic Control in Insurance, Springer-Verlag, London, 2008.
12. H. Schmidli, On capital injections and dividends with tax in a classical risk model, Insurance Math. Econom. 71 (2016), 138-144.
13. H. Schmidli, On capital injections and dividends with tax in a diusion approximation, Scand. Actuarial J. (2016), forthcoming.
14. S. E. Shreve, J. P. Lehoczky, and D. P. Gaver, Optimal consumption for general diusions with absorbing and re
ecting barriers, SIAM J. Control Optim. 22 (1984), 55-75.