A Journal "Theory of Probability and Mathematical Statistics"
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



DIVIDENDS WITH TAX AND CAPITAL INJECTION IN A SPECTRALLY NEGATIVE LEVY RISK MODEL

H. SCHMIDLI

Download PDF

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 di usion 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.

Bibliography:
1. S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scienti c, 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 in nite-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 di usion approximation, Scand. Actuarial J. (2016), forthcoming.
14. S. E. Shreve, J. P. Lehoczky, and D. P. Gaver, Optimal consumption for general di usions with absorbing and re ecting barriers, SIAM J. Control Optim. 22 (1984), 55-75.