The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion

We consider a diffusion perturbed classical compound Poisson risk model in the presence of a constant dividend barrier. An integro-differential equation with certain boundary conditions for the n-th moment of the discounted dividend payments prior to ruin is derived and solved. Its solution can be expressed in terms of the expected discounted penalty (Gerber-Shiu) functions due to oscillation in the corresponding perturbed risk model without a barrier. When the discount factor δ is zero, we show that all the results can be expressed in terms of the non-ruin probability in the perturbed risk model without a barrier.

[1]  On the moments of the surplus process perturbed by diffusion , 2002 .

[2]  Carl-Otto Segerdahl.  On some distributions in time connected with the collective theory of risk , 1970 .

[3]  Cary Chi-Liang Tsai.  On the discounted distribution functions of the surplus process perturbed by diffusion , 2001 .

[4]  Gordon E. Willmot,et al.  A generalized defective renewal equation for the surplus process perturbed by diffusion , 2002 .

[5]  José Garrido,et al.  The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion , 2005 .

[6]  The joint density function of three characteristics on jump-diffusion risk process , 2003 .

[7]  Hans U. Gerber,et al.  On the discounted penalty at ruin in a jump-diffusion and the perpetual put option , 1998 .

[8]  Howard R. Waters,et al.  Some Optimal Dividends Problems , 2004, ASTIN Bulletin.

[9]  Hansjörg Albrecher,et al.  Risk Theory with a nonlinear Dividend Barrier , 2002, Computing.

[10]  José Garrido,et al.  On a class of renewal risk models with a constant dividend barrier , 2004 .

[11]  Rong Wu,et al.  Some distributions for classical risk process that is perturbed by diffusion , 2000 .

[12]  Shuanming Li,et al.  The Diffusion Perturbed Compound Poisson RiskModel with a Dividend Barrier , 2006 .

[13]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[14]  S. Haberman,et al.  An Introduction to Mathematical Risk Theory . By Hans U. Gerber [S. S. Huebner Foundation, R. D. Irwin Inc. Homeward Illinois, 1979] , 1981 .

[15]  Cary Chi-Liang Tsai,et al.  On the expectations of the present values of the time of ruin perturbed by diffusion , 2003 .

[16]  Hans U. Gerber,et al.  Games of Economic Survival with Discrete- and Continuous-Income Processes , 1972, Oper. Res..

[17]  A decomposition of the ruin probability for the risk process perturbed by diffusion , 2001 .

[18]  Hans U. Gerber,et al.  An extension of the renewal equation and its application in the collective theory of risk , 1970 .

[19]  Bjarne Højgaard,et al.  Optimal Dynamic Premium Control in Non-life Insurance. Maximizing Dividend Pay-outs , 2002 .

[20]  Jostein Paulsen,et al.  Optimal choice of dividend barriers for a risk process with stochastic return on investments , 1997 .

[21]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[22]  Jürgen Hartinger,et al.  On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier , 2005 .

[23]  Gordon E. Willmot,et al.  The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function , 2003 .

[24]  Chuancun Yin,et al.  The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion , 2003 .

[25]  Hans U. Gerber,et al.  Optimal Dividends , 2004 .

[26]  Sidney C. Port,et al.  Brownian Motion and Classical Potential Theory , 1978 .

[27]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[28]  Hansjörg Albrecher,et al.  On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times , 2005 .

[29]  Hans U. Gerber,et al.  On the probability of ruin in the presence of a linear dividend barrier , 1981 .