具有Vasicek利率的风险过程的破产概率的分解(英文)

In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities： the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.     

随机收益环境下的破产时刻罚金折现期望

This paper considers the expected discounted penalty function Φ(u) for the perturbed compound Poisson risk model with stochastic return on investments. After presenting an integro-differential equation that the expected discounted penalty function satisfies, the paper derives the closed form solution by constructing an identical equation. The exact expression for Φ (0) is given using the Laplace transform technique when interest rate is constant. Applications of the results are given to the ruin probability and moments of the deficit at ruin.     

Positive Solutions of Boundary Value Problem for a Coupled System of Nonlinear Third-order Differential Equations

By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

Estimating the shareholder's terminal payoff based on insurer's solvency ratio in mixed fractional Brownian market

This paper studies the insurer's solvency ratio model in a class of mixed fractional Brownian motion(MFBM) market, where the prices of assets follow a Wick-It stochastic differential equation driven by the MFBM, by the method of the stochastic calculus of the MFBM and the pricing formula of European call option for the MFBM, the explicit formula for the expected present value of shareholders' terminal payoff is given. The model extends the existing results.     

Ruin Probabilities of a Surplus Process Described by PDMPs

In this paper we mainly study the ruin probability of a surplus process described by a piecewise deterministic Markov process （PDMP）. An integro-differential equation for the ruin probability is derived. Under a certain assumption, it can be transformed into the ruin probability of a risk process whose premiums depend on the current reserves. Using the same argument as that in Asmussen and Nielsen, the ruin probability and its upper bounds are obtained. Finally, we give an analytic expression for ruin probability and its upper bounds when the claim-size is exponentially distributed.     

Optimal dividend strategy in compound binomial model with bounded dividend rates

We consider the compound binomial model, and assume that dividends are paid to the shareholders according to an admissible strategy with dividend rates bounded by a constant.The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin. We show that the optimal value function is the unique solution of a discrete HJB equation. Moreover, we obtain some properties of the optimal payment strategy, and offer a simple algorithm for obtaining the optimal strategy. The key of our method is to transform the value function. Numerical examples are presented to illustrate the transformation method.     

The optimal policy for insurance company under consideration of internal competition and the time value of ruin

This paper considers the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside the company. The objective is to find a reinsurance policy and a dividend payment scheme so as to maximize the expected discounted value of the dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the corresponding constrained Hamilton-Jacobi-Bellman （HJB） equation, we obtain the value function and the optimal reinsurance policy and dividend payment.     

Constant barrier strategies in a two-state Markov-modulated dual risk model

In this paper,we consider the dividend problem in a two-state Markov-modulated dual risk model,in which the gain arrivals,gain sizes and expenses are influenced by a Markov process.A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived.In the case of exponential gain sizes,the equations are solved and the best barrier is obtained via numerical example.Finally,using numerical example,we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model.Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.     

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.     

常利率下Cox风险过程的罚金折现期望函数

本文考虑了常利率环境下Cox风险模型的罚金折现期望值，利用后向差分法，得到了条件期望值与平稳情形时的期望值分别所满足的积分方程．并且，给出了一个强度过程为二状态马尔可夫过程及索赔服从指数分布的例子．     

The compound Poisson process perturbed by a diffusion with a threshold dividend strategy

In this paper, we consider the compound Poisson process perturbed by a diffusion in the presence of the so‐called threshold dividend strategy. Within this framework, we prove the twice continuous differentiability of the expected discounted value of all dividends until ruin. We also derive integro‐differential equations for the expected discounted value of all dividends until ruin and obtain explicit expressions for the solution to the equations. Along the same line, we establish explicit expressions for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin. In the case of exponential claims, some examples are provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal choice of dividend barriers for a risk process with stochastic return on investments

We consider a risk process with stochastic return on investments and we are interested in expected present value of all dividends paid until ruin occurs when the company uses a simple barrier strategy, i.e. when it pays dividends whenever its surplus reaches a level b. It is shown that given the barrier b, this expected value can be found by solving a boundary value problem for an integro-differential equation. The solution is then found in two special cases; when return on investments is constant and the surplus generating process is compound Poisson with exponentially distributed claims, and also when both return on investments as well as the surplus generating process are Brownian motions with drift. Also in this latter case we are able to find the optimal barrier b^*, i.e. the barrier that gives the highest expected present value of dividends. Parallell with this we treat the problem of finding the Laplace transform of the distribution of the time to ruin when a barrier strategy is employed, noting that the probability of eventual ruin is 1 in this case. The paper ends with a short discussion of the same problems when a time dependent barrier is employed.     

On the expected discounted penalty function for risk process with tax

In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.     

一类随机利率下的破产时罚金折现期望

本文在经典风险模型下, 引进带有一种随机利率的破产时罚金折现期望的概念, 其利率的随机性通过标准Wiener过程和Poisson过程来描述. 给出破产时罚金折现期望所满足的更新方程, 并利用这个更新方程给出破产时罚金折现期望的渐近公式.     

随机保费风险模型下的平均折现罚金函数

本文研究随机保费风险模型下与破产时刻相关的平均折现罚金函数. 与经典的Cram\'{e}r-Lundberg模型相比这里的保费过程不再是时间的线性函数, 而是一个与理赔独立的复合Possion过程. 我们得到了罚金函数所满足的积分方程, 它提供了一种研究破产量的统一方法. 利用该积分方程我们得到了破产时刻, 破产时赤字, 破产前瞬时盈余的Laplace变换; 并在指数分布的特殊情况下求出了他们的显著表达式, 推广了Boikov (2003)的结论.     

变保费率Cox风险模型的研究

研究了当保费率随理赔强度的变化而变化时C ox风险模型的折现罚金函数,利用后向差分法得到了折现罚金函数所满足的积分方程,进而得到了破产概率,破产前瞬时盈余、破产时赤字的各阶矩所满足的积分方程.最后给出当理赔额服从指数分布,理赔强度为两状态的马氏过程时破产概率的拉普拉斯变换,对一些具体数值计算出了破产概率的表达式.