Differentiability of dividends function on jump-diffusion risk process with a barrier dividend strategy

Abstract

We consider a dividends model with a stochastic jump perturbed by diffusion. First, we prove that the expected discounted dividends function is twice continuously differentiable under the condition that the claim distribution function has continuous density. Then we show that the expected discounted dividends function under a barrier strategy satisfies some integro-differential equation of defective renewal type, and the solution of which can be explicitly expressed as a convolution formula. Finally, we study the Laplace transform of ruin time on the modified surplus process.

Keywords

Expected discounted dividends / ruin time / integro-differential equation / Laplace transform / barrier strategy

References

[1] Asmussen S. Ruin Probabilities. Singapore: World Scientific, 2000

[2] Cai J, Dickson D C M. Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance Math Econom, 2003, 32(1): 61-71

[3] De Finetti B. Su un’ impostazione alternativa dell teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries, 1957, 2: 433-443

[4] Dufresne F, Gerber H U. Risk theory for the compound Poisson process that perturbed by diffusion. Insurance Math Econom, 1991, 10(1): 51-59

[5] Gao H, Yin C. A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy. Appl Math Comput, 2008, 205: 454-464

[6] Gerber H U. An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift, 1970, 205-210

[7] Gerber H U, Shiu E S W. Optimal dividends: Analysis with Brownian motion. North Amer Actuarial J, 2004, 8(1): 1-20

[8] Mikhlin S G. Integral Equations. London: Pergamon Press, 1957

[9] Paulsen J, Gjessing H K. Ruin theory with stochastic return on investments. Adv Appl Probab, 1997, 29: 965-985

[10] Revuz D, Yor M. Continuous Martingales and Brownian Motion. Berlin-New York: Springer-Verlag, 1991

[11] Rolski T, Schmidli H, Teugels J. Stochastic Processes for Insurance and Finance. Chichester-New York: John Wiley and Sons, 1999

[12] Wan N. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. Insurance Math Econom, 2007, 40(3): 509-523

[13] Wu R, Wang G J, Wei L. Joint distributions of some actuarial random vectors containing the time of ruin. Insurance Math Econom, 2003, 33(1): 147-161

[14] Yuen K C, Lu Y H, Wu R. The compound Poisson process perturbed by a diffusion with a threshold dividend strategy. Appl Stoch Models Bus Ind, 2009, 25(1): 73-93

[15] Zhang C S, Wang G J. The joint density function of three characteristics on jumpdiffusion risk process. Insurance Math Econom, 2003, 32(3): 445-455

[16] Zhang C S, Wu R. Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion. J Appl Probab, 2002, 39: 517-532

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg