In this paper we consider the risk process described by a piecewise deterministic Markov processes (PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integrodi differ...In this paper we consider the risk process described by a piecewise deterministic Markov processes (PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integrodi differential equation satisfied by this distribution. We obtain the explicit expressions for it for certain choices of the claim amount distribution.展开更多
A dual model of the perturbed classical compound Poisson risk model is considered under a constant dividend barrier. A new method is used in deriving the boundary condition of the equation for the expectation function...A dual model of the perturbed classical compound Poisson risk model is considered under a constant dividend barrier. A new method is used in deriving the boundary condition of the equation for the expectation function by studying the local time of a related process. We obtain the expression for the expected discount dividend function in terms of those in the corresponding perturbed compound Poisson risk model without barriers. A special case in which the gain size is phase-type distributed is illustrated. We also consider the existence of the optimal dividend level.展开更多
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
With the ever-evolving of modern risk theory,more and more attention should be paid to the modification of the classical risk theory. In this paper a risk process with premiums dependent on the current reserve is cons...With the ever-evolving of modern risk theory,more and more attention should be paid to the modification of the classical risk theory. In this paper a risk process with premiums dependent on the current reserve is considered. An explicit expression for the joint distribution of the time of ruin,the surplus immediately before ruin and the deficit at ruin is derived. Finally,some important actuarial diagnostics including the ultimate ruin probability is investigated.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
We consider the discrete risk model with exponential claim sizes. We derive the finite explicit elementary expression for the joint density function of three characteristics： the time of ruin, the surplus immediately...We consider the discrete risk model with exponential claim sizes. We derive the finite explicit elementary expression for the joint density function of three characteristics： the time of ruin, the surplus immediately before ruin, and the deficit at ruin. By using the explicit joint density function, we give a concise expression for the Gerber-Shiu function with no dividends. FinMly, we obtain an integral equation for the Gerber-Shiu function under the barrier dividend strategy. The solution can be expressed as a combination of the Gerber-Shiu function without dividends and the solution of the corresponding homogeneous integral equation. This latter function is given clearly by means of the Gerber- Shiu function without dividends .展开更多
基金the National Natural Science Foundation of China (Grant No. 10271087)National Science Foundation of Jiangsu education Ministry (Grant No. 02KJB110002).the National Natural Science Foundation of China (Grant No. 10271062)the Research Fund for th
文摘In this paper we consider the risk process described by a piecewise deterministic Markov processes (PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integrodi differential equation satisfied by this distribution. We obtain the explicit expressions for it for certain choices of the claim amount distribution.
基金the National Basic Research Program of China (973 Program)(No.2007CB814905)the National Natural Science Foundation of China (No.10571092)the Research Fund of the Doctorial Program of Higher Education
文摘A dual model of the perturbed classical compound Poisson risk model is considered under a constant dividend barrier. A new method is used in deriving the boundary condition of the equation for the expectation function by studying the local time of a related process. We obtain the expression for the expected discount dividend function in terms of those in the corresponding perturbed compound Poisson risk model without barriers. A special case in which the gain size is phase-type distributed is illustrated. We also consider the existence of the optimal dividend level.
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.
基金Ministry of Education in China(MOE)Youth Projects of Humanities and Social Sciences(Nos.14YJCZH048,15YJCZH204)National Natural Science Foundations of China(Nos.11401436,11601382,11101434,11571372)+2 种基金National Social Science Foundation of China(No.15BJY122)Hunan Provincial Natural Science Foundation of China(No.13JJ5043)Mathematics and Interdisciplinary Sciences Project,Central South University
文摘With the ever-evolving of modern risk theory,more and more attention should be paid to the modification of the classical risk theory. In this paper a risk process with premiums dependent on the current reserve is considered. An explicit expression for the joint distribution of the time of ruin,the surplus immediately before ruin and the deficit at ruin is derived. Finally,some important actuarial diagnostics including the ultimate ruin probability is investigated.
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.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Crant Nos. 11226203, 11226204, 11171164, 11271385, 11401436）.
文摘We consider the discrete risk model with exponential claim sizes. We derive the finite explicit elementary expression for the joint density function of three characteristics： the time of ruin, the surplus immediately before ruin, and the deficit at ruin. By using the explicit joint density function, we give a concise expression for the Gerber-Shiu function with no dividends. FinMly, we obtain an integral equation for the Gerber-Shiu function under the barrier dividend strategy. The solution can be expressed as a combination of the Gerber-Shiu function without dividends and the solution of the corresponding homogeneous integral equation. This latter function is given clearly by means of the Gerber- Shiu function without dividends .
