Research on Ruin Problems with Two-type by-claims and Delay Period Based on Compound Binomial Risk Model

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution

In this paper; survival (non-ruin) probability after a definite time period of an insurance company is studied in a discrete time model based on nonhomogenous claim occurrences. Furthermore, distributions of the minimum and maximum levels of surplus in compound binomial risk model with nonhomogeneous claim occurrences are obtained and some of its characteristics are given.

This study examines the implementation of the ruin probability model in risk management in life insurance companies. The main focus of this study is to evaluate how factors such as initial surplus, premium revenue level, and claim frequency affect the ruin probability of insurance companies. Using the collective risk model approach and relevant claim distribution, this study develops two methods to calculate the ruin probability: an analytical approach and a Monte Carlo simulation. The simulation results show that increasing the initial surplus and premium level significantly reduces the ruin risk, while increasing the claim frequency increases the ruin probability. In addition, the gamma claim distribution is more suitable for modeling claims in life insurance than the exponential distribution. Model validation is carried out by comparing the prediction results with historical data of insurance companies, which shows a high level of accuracy. This study provides important insights for insurance companies in designing more effective and optimal risk management strategies

Asymptotics for a time-dependent renewal risk model with subexponential main claims and delayed claims

In the classical risk model with Poisson arrivals, we study a functional approach which can be used to obtain new approximation formulae for the probability of ultimate ruin. In particular, we consider a map Φ between appropriate function spaces with Φ(f) = ψ, where f denotes the density of claim sizes in the model and ψ is the function that gives the associated infinite-time ruin probability, ψ(x), for a given initial surplus x. Suppose then that , represent two claim size densities in the model, with corresponding outputs ψ(0) and ψ(1) respectively, and that only ψ(0) is known analytically. By establishing that the map Φ has derivatives of any order, we can approximate ψ(1) by using ψ(0) and the derivatives of Φ, provided that are not “too far” from each other. Since the explicit formulae for high order derivatives of Φ are typically complicated, we explain how computer algebra can be used for the implementation of the method. We also discuss how the functional approach can be used to obtain approximations for other quantities of interest in risk theory and we carry out some numerical illustrations to investigate the accuracy of the approximations.

Let \(R(t) = u + ct - \sum\limits_{i = 1}^{N(t)} {X_i ,t \geqslant 0} \) be the renewal risk model, with FX(x) being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of FX(x) ∈ S*(γ, γ ⩾ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ(u, u + z] for every 0 < z < ∞. On one hand, the asymptotic behavior of ψ(u) can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.

of thesis entitled STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES Submitted by CHEN YIQING for the degree of Doctor of Philosophy at The University of Hong Kong in August 2009 In recent years, insurance risk studies in the presence of dependence structure and heavy-tailed claims are popular topics in actuarial science. In this thesis, we consider the renewal risk model with certain dependence structures, namely negative dependence and quasi-asymptotic independence. We also assume that claim sizes follow a heavy-tailed distribution, in particular, a subexponential distribution. We focus on studying the impact of heavy tails and dependence structures on ruin probabilities and the tail probabilities of aggregate claims. The asymptotic behavior of the ultimate ruin probability in the renewal risk model was studied in the presence of negatively dependent claim sizes and a constant force of interest. A purely probabilistic approach was employed to establish a simple asymptotic formula for the ruin probability. This result coincides with some known results in the recent literature, and discovers a phenomenon that heavy tails can offset the impact of the dependence structure of claim sizes on the ruin probability. A two-dimensional renewal risk model was considered in which the insurance company consists of two dependent classes of business. The dependence between the two classes comes from the fact that their claims arrive according to a common renewal counting process. For this model, several ruin probabilities were defined and simple asymptotic formulae for these ruin probabilities were derived. In particular, the obtained asymptotic formulae hold uniformly for all time horizons. This uniformity greatly enhances the theoretical value of the results. The next problem was to study precise estimates of the large deviation probabilities of aggregate claims with sizes which are negatively dependent. Specially, the problem for the so-called quasi renewal risk model was investigated in which inter-arrival times are also negatively dependent. A new dependence structure was considered and was termed “quasi-asymptotic independence” structure, which is more general and verifiable than some commonlyused dependence structures. The proposed dependence structure covers a wide range of positive and negative dependence. For the case of Pareto-like tails, the tail probability of the sum was proved to be asymptotically equivalent to the sum of tail probabilities. This confirms again that heavy tails offset the impact of dependence between summands. This result was then extended to randomly weighted sums and random sums. All the obtained results have obvious applications to concrete actuarial problems. In the last part of the thesis, each primary random variable representing the net loss during a period was associated with a random weight or a stochastic discount factor. Therefore, the weighted sum represents the stochastic present value of aggregate losses. Suppose that the primary random variables are long tailed, the weights are positive and bounded, and the primary random variables and their weights are independent. The tail probability of the maximum of the weighted sums was shown to be asymptotically equivalent to that of the last weighted sum. STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES

According to the characteristics of risk model with perturbed by diffusion, a new kind of stochastic process— F-B process is proposed following from Fuzzy Markov Skeleton process. Then the ruin probabilities (caused by oscillation or by a claim) are discussed by F-B process. We derive a distribution of the ruin probability and a distribution of the surplus immediately before the time of ruin. Then continuous model and discrete model, risk model with compound Poisson process and risk model with double Poisson process, are used to illustrate the implementation of the model. Next limit theorems of two continuous risk models perturbed by diffusion are discussed. Finally simulation experiment is introduced.

Asymptotic ruin probability of a renewal risk model with dependent by-claims and stochastic returns

Markovian risk process

In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

On the time value of absolute ruin with tax

In this paper the probability of ruin is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model. The idea is originally enlightened by Jasiulewicz (2001). We make a slight modification on the model and a generalization on the intensity process. By a “backward differential argument” and the Markov property of the intensity process we strictly derive the integral equation satisfied by the probability of ruin. Further, we solve the equation when the intensity process is a homogeneous n-state Markov process by Laplace transforms. At the end of the paper, an example is given.

this paper discusses ruin problems in the renewal risk model with interest force. By using inductive method, the recursive expressions of the distribution of the ruin probability, the distribution of maximum surplus before the ruin and distribution of minimum surplus before the ruin are obtained, and the corresponding integral equation for the distributions are obtained. Keywords-risk model; Interest force; renewal risk model; ruin probability; maximum surplus; minimum surplus