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Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. (English) Zbl 1183.91077
Summary: In the absence of dividends, the surplus of an insurance company is modeled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed-form solutions are given. Accordingly, the Gerber-Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed.

91B30Risk theory, insurance
45J05Integro-ordinary differential equations
60K05Renewal theory
60K10Applications of renewal theory
Full Text: DOI
[1] Asmussen, S.; Taksar, M.: Controlled diffusion models for optimal dividend pay-out. Insurance: mathematics and economics 20, 1-15 (1997) · Zbl 1065.91529
[2] Chiu, S. N.; Yin, C. C.: The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: mathematics and economics 33, No. 1, 59-66 (2003) · Zbl 1055.91042
[3] De Finetti, B.: Su un’impostazione alternativa dell teoria colletiva del rischio. Transactions of the XV international congress of actuaries 2, 433-443 (1957)
[4] Dufresne, F.; Gerber, H. U.: Risk theory for the command Poisson process that is perturbed by diffusion. Insurance: mathematics and economics 10, 51-59 (1991) · Zbl 0723.62065
[5] Gerber, H. U.: An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk aktuarietidskrift, 205-210 (1970) · Zbl 0229.60062
[6] Gerber, H. U.; Landry, B.: On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: mathematics and economics 22, 263-276 (1998) · Zbl 0924.60075
[7] Gerber, H. U.; Shiu, E. S. W.: Optimal dividends: analysis with Brownian motion. North American actuarial journal 8, No. 1, 1-20 (2004) · Zbl 1085.62122
[8] Gerber, H. U.; Shiu, E. S. W.: On optimal dividend strategy in the compound Poisson model. North American actuarial journal 10, No. 2, 76-93 (2005)
[9] Gerber, H. U.; Shiu, E. S. W.: On optimal dividends: from reflection to refraction. Journal of computational and applied mathematics 186, 4-22 (2006) · Zbl 1089.91023
[10] Li, S., 2005. The distribution of the dividend payment in the compound Poisson risk model perturbed by diffusion. Working paper
[11] Lin, X. S.; Willmot, G. E.; Drekic, S.: The classical risk model with a constant dividend barrier. Insurance: mathematics and economics 33, 551-566 (2003) · Zbl 1103.91369
[12] Lin, X. S.; Pavlova, K. P.: The compound Poisson risk model with a threshold dividend strategy. Insurance: mathematics and economics 38, 57-80 (2006) · Zbl 1157.91383
[13] Luo, J., 2005. The pricing theory and the application of the American options and the numerical algorithm of the implied volatility. Doctoral dissertation
[14] Jeanblanc-Picqué, M.; Shiryaev, A. N.: Optimization of the flow of dividends. Russian mathematical surveys 20, 257-277 (1995) · Zbl 0878.90014
[15] Wang, G.; Wu, R.: Some distributions for classical risk processes that is perturbed by diffusion. Insurance: mathematics and economics 26, 15-24 (2000) · Zbl 0961.62095