Power identities for L\'evy risk models under taxation and capital injections
In this paper we study a spectrally negative L\'evy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction. It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided.
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- Kulenko, Natalie & Schmidli, Hanspeter, 2008. "Optimal dividend strategies in a Cramér-Lundberg model with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 270-278, October.
- Nie, Ciyu & Dickson, David C. M. & Li, Shuanming, 2011. "Minimizing the ruin probability through capital injections," Annals of Actuarial Science, Cambridge University Press, vol. 5(02), pages 195-209, September.
- Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 34(01), pages 49-74, May.
- Albrecher, Hansjörg & Lautscham, Volkmar, 2013. "From Ruin to Bankruptcy for Compound Poisson Surplus Processes," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 43(02), pages 213-243, May.
- Albrecher, Hansjörg & Borst, Sem & Boxma, Onno & Resing, Jacques, 2009. "The tax identity in risk theory -- a simple proof and an extension," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 304-306, April.
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