On the dual risk model with tax payments. (English) Zbl 1141.91481

Summary: We study the dual risk process in ruin theory [see e.g. H. Cramér, Collective risk theory: a survey of the theory from the point of view of the theory of stochastic processes. Stockholm: AB Nordiska Bokhandeln (1955); L. Takács [Combinatorial methods in the theory of stochastic processes. New York etc.: John Wiley (1967; Zbl 0162.21303) and B. Avanzi et al., Insur. Math. Econ. 41, No. 1, 111–123 (2007; Zbl 1131.91026)] in the presence of tax payments according to a loss-carry forward system. For arbitrary inter-innovation time distributions and exponentially distributed innovation sizes, an expression for the ruin probability with tax is obtained in terms of the ruin probability without taxation. Furthermore, expressions for the Laplace transform of the time to ruin and arbitrary moments of discounted tax payments in terms of passage times of the risk process are determined. Under the assumption that the inter-innovation times are (mixtures of) exponentials, explicit expressions are obtained. Finally, we determine the critical surplus level at which it is optimal for the tax authority to start collecting tax payments.


91B30 Risk theory, insurance (MSC2010)
Full Text: DOI


[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), National Bureau of Standards: National Bureau of Standards Washington · Zbl 0515.33001
[2] Ahn, S.; Badescu, A.; Ramaswami, V., Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier, Queueing Systems, 55, 4, 207-222 (2007) · Zbl 1124.60067
[3] Albrecher, H.; Hipp, C., Lundberg’s risk process with tax, Blätter der DGVFM, 28, 1, 13-28 (2007) · Zbl 1119.62103
[4] Avanzi, B.; Gerber, H. U.; Shiu, E. S.W., Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41, 111-123 (2007) · Zbl 1131.91026
[5] Badescu, A. L.; Drekic, S.; Landriault, D., Analysis of a threshold dividend strategy for a MAP risk model, Scandinavian Actuarial Journal, 4, 248-260 (2007) · Zbl 1164.91025
[6] Bühlmann, H., Mathematical Methods in Risk Theory (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0209.23302
[7] Cohen, J. W., The Single Server Queue (1982), North-Holland: North-Holland Amsterdam · Zbl 0481.60003
[8] Cramér, H., Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes (1955), Ab Nordiska Bokhandeln: Ab Nordiska Bokhandeln Stockholm
[9] Gerber, H. U., An Introduction to Mathematical Risk Theory (1979), S.S. Huebner Foundation: S.S. Huebner Foundation Philadelphia · Zbl 0431.62066
[10] Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics: A Foundation for Computer Science (1994), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0836.00001
[11] Grandell, J., Aspects of Risk Theory (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0717.62100
[12] Prabhu, N. U., Stochastic Storage Processes (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0453.60094
[13] Ramaswami, V., Passage times in fluid models with application to risk processes, Methodology and Computing in Applied Probability, 8, 4, 497-515 (2006) · Zbl 1110.60067
[14] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic Processes for Insurance and Finance (1999), Wiley: Wiley New York · Zbl 0940.60005
[15] Takacs, L., Combinatorial methods in the Theory of Stochastic Processes (1967), Wiley: Wiley New York · Zbl 0189.17602
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