THE JOINT DISTRIBUTIONS OF SOME ACTUARIAL DIAGNOSTICS FOR THE JUMP-DIFFUSION RISK PROCESS

In this article, the joint distributions of several actuarial diagnostics which are important to insurers＇ running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus process leaving zero ultimately （simply, the ultimately leaving-time）, the surplus immediately prior to ruin, the supreme profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. The explicit expressions for their distributions are obtained mainly by the various properties of Levy process, such as the homogeneous strong Markov property and the spatial homogeneity property etc, moveover, the many properties for Brownian motion.

EXPLICIT EXPRESSIONS FOR SOME DISTRIBUTIONS RELATED TO RUIN PROBLEMS

The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim amount distribution is a finite mixture of exponential distributions or a Gamma (2, α) distribution.

OPTIMAL PROPORTIONAL REINSURANCE WITH CONSTANT DIVIDEND BARRIER

In this article, we consider an optimal proportional reinsurance with constant dividend barrier. First, we derive the Hamilton-Jacobi-Bellman equation satisfied by the expected discounted dividend payment, and then get the optimal stochastic control and the optimal constant barrier. Secondly, under the optimal constant dividend barrier strategy, we consider the moments of the discounted dividend payment and their explicit expressions are given. Finally, we discuss the Laplace transform of the time of ruin and its explicit expression is also given.

ASYMPTOTIC THEORY FOR A RISK PROCESS WITH A HIGH DIVIDEND BARRIER

A modified classical model with a dividend barrier is considered. It is shown that there is a simple approximation formula for the time of ruin when the level of dividend barrier is high and the claim sizes have a distribution that belongs to S（γ） with γ 〉0.

The dividend function in the jump-diffusion dual model withbarrier dividend strategy

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.

Joint Distribution for the Risk Process with Premiums Depending on the Current Reserve

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.

Gerber-Shiu function of a discrete risk model with and without a constant dividend barrier

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 .

The Gerber-Shiu Expected Discounted Penalty Function for Lévy Insurance Risk Processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.

Some Results for Classical Risk Process with Stochastic Return on Investments

In this paper, we discuss the classical risk process with stochastic return on investment. We prove some properties of the ruin probability, the supremum distribution before ruin and the surplus distribution at the time of ruin and derive the integro-differential equations satisfied by these distributions respectively.