In this article, the expected discounted penalty function Фδ,α （u） with constant interest δ and ＂discounted factor＂ exp（-αTδ） is considered. As a result, the integral equation of Фδ,α （u） is derived a...In this article, the expected discounted penalty function Фδ,α （u） with constant interest δ and ＂discounted factor＂ exp（-αTδ） is considered. As a result, the integral equation of Фδ,α （u） is derived and an exact solution for Фδ,α （0） is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.展开更多
In this paper, we consider a general expression for Ф（u, x, y）, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is u. In the renewal risk model, this density...In this paper, we consider a general expression for Ф（u, x, y）, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is u. In the renewal risk model, this density function is expressed in terms of the corresponding density function when the initial surplus is O. In the compound Poisson risk process with phase-type claim size, we derive an explicit expression for Ф（u, x, y）. Finally, we give a numerical example to illustrate the application of these results.展开更多
In this paper, we consider a Gerber-Shiu discounted penalty function in Sparre Andersen risk process in which claim inter-arrival times have a phase-type （2） distribution, a distribution with a density satisfying a ...In this paper, we consider a Gerber-Shiu discounted penalty function in Sparre Andersen risk process in which claim inter-arrival times have a phase-type （2） distribution, a distribution with a density satisfying a second order linear differential equation. By conditioning on the time and the amount of the first claim, we derive a Laplace transform of the Gerber-Shiu discounted penalty function, and then we consider the joint density function of the surplus prior to ruin and the deficit at ruin and some ruin related problems. Finally, we give a numerical example to illustrate the application of the results.展开更多
文摘In this article, the expected discounted penalty function Фδ,α （u） with constant interest δ and ＂discounted factor＂ exp（-αTδ） is considered. As a result, the integral equation of Фδ,α （u） is derived and an exact solution for Фδ,α （0） is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.
文摘In this paper, we consider a general expression for Ф（u, x, y）, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is u. In the renewal risk model, this density function is expressed in terms of the corresponding density function when the initial surplus is O. In the compound Poisson risk process with phase-type claim size, we derive an explicit expression for Ф（u, x, y）. Finally, we give a numerical example to illustrate the application of these results.
文摘In this paper, we consider a Gerber-Shiu discounted penalty function in Sparre Andersen risk process in which claim inter-arrival times have a phase-type （2） distribution, a distribution with a density satisfying a second order linear differential equation. By conditioning on the time and the amount of the first claim, we derive a Laplace transform of the Gerber-Shiu discounted penalty function, and then we consider the joint density function of the surplus prior to ruin and the deficit at ruin and some ruin related problems. Finally, we give a numerical example to illustrate the application of the results.
