Ruin probabilities for a risk process with stochastic return on investments

In this paper, we consider a risk process with stochastic return on investments. The basic risk process is the classical risk process while the return on the investment generating process is a compound Poisson process plus a Brownian motion with positive drift. We obtain an integral equation for the ultimate ruin probability which is twice continuously differentiable under certain conditions. We then derive explicit expressions for the lower bound for the ruin probability. We also study a joint distribution related to exponential functionals of Brownian motion which is required in the derivations of the explicit expressions for the lower bound.     

一类Cox风险过程中的三联分布(英文)

本文研究了一类Cox风险过程破产时、破产瞬间前的余额、破产时的赤字这三个重要精算量的联合分布,并给出了一些密度测度的分布.     

带有随机干扰和确定投资回报风险过程下的一些分布

我们考虑既带有随机干扰又带有确定投资回报的风险过程, 得到了破产前瞬间盈余的分布$F_{\delta}(u,x)$及破产前瞬间盈余和破产时赤字的联合分布$H_{\delta}(u,x,y)$所满足的积分表达, 连续性及二次连续可微性和积分--微分方程. 同时, 只有随机干扰的风险模型下的破产前瞬间盈余的分布及破产前瞬间盈余和破产时赤字的联合分布所满足的性质也被得到. 已有文献中的诸多有关结果均可以通过令我们结论中的某些参数特殊化为零而得到.     

最佳投资策略下投资额函数的渐近性质

本文考虑一个经典风险模型,且允许保险公司投资股票市场,通过选择适当的投资策略使破产概率达到最小,并求出当分布函数F(x)是正则变化函数时,投资额函数A(x)的近似表达式.     

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.     

Analysis of risk models using a level crossing technique

This paper analyzes ruin-like risk models in Insurance, which are variants of the Cramer–Lundberg (C–L) model with a barrier or a threshold. We consider three model variants, which have different portfolio strategies when the risk reserve reaches the barrier or exceeds the threshold. In these models we construct a time-extended risk process defined on cycles of a specific renewal process. The time until ruin is equal to one cycle of the specific renewal process. We also consider a fourth model, which is a variant of a model proposed by Dickson and Waters (2004). The analysis of each model employs a level crossing method (LC) to derive the steady-state probability distribution of the time-extended risk process. From the derived distribution we compute the expected time until ruin, the probability distribution of the deficit at ruin, and related quantities of interest.     

On the distribution tail of an integrated risk model: A numerical approach

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of a risky asset and a riskless asset. The stock price is modelled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.     

A risk model with paying dividends and random environment

We consider a discrete time risk model where dividends are paid to insureds and the claim size has a discrete phase-type distribution, but the claim sizes vary according to an underlying Markov process called an environment process. In addition, the probability of paying the next dividend is affected by the current state of the underlying Markov process. We provide explicit expressions for the ruin probability and the deficit distribution at ruin by extracting a QBD (quasi-birth-and-death) structure in the model and then analyzing the QBD process. Numerical examples are also given.     

Analysis of risk models using a level crossing technique

This paper analyzes ruin-like risk models in Insurance, which are variants of the Cramer–Lundberg (C–L) model with a barrier or a threshold. We consider three model variants, which have different portfolio strategies when the risk reserve reaches the barrier or exceeds the threshold. In these models we construct a time-extended risk process defined on cycles of a specific renewal process. The time until ruin is equal to one cycle of the specific renewal process. We also consider a fourth model, which is a variant of a model proposed by Dickson and Waters (2004). The analysis of each model employs a level crossing method (LC) to derive the steady-state probability distribution of the time-extended risk process. From the derived distribution we compute the expected time until ruin, the probability distribution of the deficit at ruin, and related quantities of interest.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

Risk aggregation based on the Poisson INAR(1) process with periodic structure

In this paper, we consider a risk model by introducing a temporal dependence between the claim numbers under periodic environment, which generalizes several discrete-time risk models. The model proposed is based on the Poisson INAR(1) process with periodic structure. We study the moment-generating function of the aggregate claims. The distribution of the aggregate claims is discussed when the individual claim size is exponentially distributed.     

Optimal investment policy of an insurance firm

In this paper we consider an investment problem by an insurance firm. As in the classical model of collective risk, it is assumed that premium payments are received deterministically from policyholders at a constant rate, while the claim process is determined by a compound Poisson process. We introduce a conversion mechanism of funds from cash into investments and vice versa. Contrary to the conventional collective risk model we do not assume a ruin barrier. Instead we introduce conversion costs to account for the problems implicit in reaching the zero boundary. The objective of the firm is to maximize its net profit by selecting an appropriate investment strategy. A diffusion approximation is suggested in order to obtain tractable results for a general claim size distribution.     

Risk process with stochastic income and two-step premium rate

In this paper we deal with the risk reserve process with stochastic premium function. We assume that the premiums sizes have exponential distribution with the rate depending on some threshold level. The representation for the discounted defective joint density of surplus and deficit at ruin is obtained.     

Markov-Modulated风险模型的极大值、极小值及零点数的联合分布

本文基于保险公司在首次破产后仍能继续运转的情形,讨论并得到了Markov-modulated风险模型中关于末离零点前盈余过程极大值、极小值及零点数的联合分布.     

Asymptotic analysis of a risk process with high dividend barrier

In this paper we study a risk model with constant high dividend barrier. We apply Keilson’s (1966) results to the asymptotic distribution of the time until occurrence of a rare event in a regenerative process, and then results of the cycle maxima for random walk to obtain the asymptotic distribution of the time to ruin and the amount of dividends paid until ruin.     

Erlang(2)过程的风险分析与美式看跌期权

在经典的风险理论中涉及到的索赔风险是服从复合Poission过程的, 与之不同, 我们考虑Erlang(2)风险过程\bd Erlang(2)分布往往见诸于控制理论中, 这里它作为索赔发生间隔时间的分布被引入了\bd 本文中, 我们介绍一个与破产时刻、破产前时刻的盈余以及破产时刻赤字有关的辅助函数$\phi(\cdot)$, 函数中涉及的这三个变量对风险模型的研究都是最基本也是最重要的\bdWillmot and Lin (1999)曾在古典连续时间风险模型之中研讨过这一函数\bd受Gerber and Shi(1997)及Willmot and Lin (2000)在古典模型下的研究过程的启发, 本文的一个重要结果就是找到破产前时刻的盈余以及破产时刻赤字的联合分布密度函数\bd 更得益于Gerber and Landry (1998)及Gerber and Shiu (1999)的思想, 我们应用以上的结果去寻求基础资产服从一定风险资产价格过程的美式看跌期权最优交易策略.     

Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.     

竞争型二元风险模型破产概率

本文研究了竞争型的二元风险模型,定义了两类破产概率以及状态过程,利用经典风险模型的已有结果和条件期望的性质,得到两类破产概率表达式,以及单个保险公司有限时间破产概率和最终破产概率,并给出两个保险公司的状态过程的概率分布列.     

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 runi probability and the surplus distribution of the risk process at the time of runi are obtained when the claim amount distribution is a finite mixture of exponential distributions of a Gamma (2,α） distribution.     

带干扰phaes-type风险模型中的最大盈余及相关分布

本文考虑了索赔时间间距为phase-type分布时带干扰更新风险模型中的破产前最大盈余、破产后赤字的分布,建立了相应的积分-微分方程.最后,讨论了当索赔时间间距为Erlang(2)分布且索赔量满足指数分布时的特殊情形.