A note on a by-claim risk model: Asymptotic results

In this note, we restudy a by-claim risk model with general dependence structures between each main claim and its by-claim. Within the framework of regular variation, we derive some asymptotic expansions for the infinite-time and finite-time ruin probabilities.     

Ruin Probabilities for the Perturbed Compound Poisson Risk Process with Investment

In this article, we consider the perturbed compound Poisson risk process with investment incomes. The risk reserve process is perturbed by an independent Brownian motion and the surplus is invested at a constant force of interest. We investigate the asymptotic behavior of the ruin probability as the initial reserve goes to infinity. Bounds and time-dependent bounds are derived for the ultimate ruin probability and the probabilities of ruin within finite time, respectively. We also obtain an explicit expression for the Laplace transform of the ultimate ruin probability.     

The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims

This article studies a continuous-time bidimensional risk model, in which an insurer simultaneously confronts two kinds of claim sharing a common renewal claim-number process. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution with extended regularly varying margins, we derive an explicit asymptotic formula for the corresponding infinite-time ruin probability.     

Gambler’s Ruin: The Duration of Play

We study the gambler’s ruin problem with a general distribution of the payoffs in each game. Assuming the expected value of the payoff distribution is negative, so that eventual ruin occurs with probability 1, we are interested in the distribution of the duration to ruin, also known as the first-passage time distribution. A generating function for this distribution is obtained. Exact expressions for the expected value and variance of this distribution, as well as asymptotic expressions for the case of large initial wealth, are derived.     

Asymptotic ruin probabilities of a two-dimensional renewal risk model with dependent inter-arrival times

Abstract

This article mainly considers the uniform asymptotics for the finite-time ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims. In this model, the two claim-number processes are arbitrarily dependent and each of them is generated by widely orthant dependent claim inter-arrival times. Two types of ruin are studied and for each type of ruin, an asymptotic formula for the finite-time ruin probability is established. These formulae possess a certain uniformity feature in the time horizon.     

BAYESIAN ESTIMATION OF RUIN PROBABILITIES WITH A HETEROGENEOUS AND HEAVY-TAILED INSURANCE CLAIM-SIZE DISTRIBUTION

This paper describes a Bayesian approach to make inference for risk reserve processes with an unknown claim‐size distribution. A flexible model based on mixtures of Erlang distributions is proposed to approximate the special features frequently observed in insurance claim sizes, such as long tails and heterogeneity. A Bayesian density estimation approach for the claim sizes is implemented using reversible jump Markov chain Monte Carlo methods. An advantage of the considered mixture model is that it belongs to the class of phase‐type distributions, and thus explicit evaluations of the ruin probabilities are possible. Furthermore, from a statistical point of view, the parametric structure of the mixtures of the Erlang distribution offers some advantages compared with the whole over‐parametrized family of phase‐type distributions. Given the observed claim arrivals and claim sizes, we show how to estimate the ruin probabilities, as a function of the initial capital, and predictive intervals that give a measure of the uncertainty in the estimations.     

The finite-time ruin probability in two non-standard renewal risk models with constant interest rate and dependent subexponential claims

This paper considers an ordinary renewal risk model and a compound renewal risk model with constant interest rate, subexponential claims and a general premium process. We derive some asymptotic results on the finite-time ruin probabilities.     

Uniform Asymptotic Estimates for Ruin Probabilities with Exponential Lévy Process Investment Returns and Two-sided Linear Heavy-tailed Claims

This paper investigates the ruin probabilities of a renewal risk model with stochastic investment returns and dependent claim sizes. The investment is described as a portfolio of one risk-free asset and one risky asset whose price process is an exponential Lévy process. The claim sizes are assumed to follow a two-sided linear process with independent and identically distributed step sizes. When the step-size distribution is heavy tailed, the paper establishes some uniform asymptotic formulas of ruin probabilities.     

The finite-time ruin probability of a discrete-time risk model with GARCH discounted factors and dependent risks

The finite-time ruin probability of a discrete-time risk model with dependent stochastic discount factors and dependent insurance and financial risks is investigated in this paper. Assume that the stochastic discount factors follow a GARCH process and the one-period insurance and financial risks form a sequence of independent and identically distributed random pairs, which are the copies of a random pair with a bivariate Sarmanov dependent distribution. When the common distribution of claim-sizes is heavy-tailed, we establish an asymptotic estimate for the finite-time ruin probability. Applying the result to a special case, we also get conservative asymptotic bounds. A numerical simulation is given at the end of the paper.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

On a discrete interaction risk model with delayed claims and stochastic incomes under random discount rates

In this paper, we study a discrete interaction risk model with delayed claims and stochastic incomes in the framework of the compound binomial model. A generalized Gerber-Shiu discounted penalty function is proposed to analyse this risk model in which the interest rates follow a Markov chain with finite state space. We derive an explicit expression for the generating function of this Gerber-Shiu discounted penalty function. Furthermore, we derive a recursive formula and a defective renewal equation for the original Gerber-Shiu discounted penalty function. As an application, the joint distributions of the surplus one period prior to ruin and the deficit at ruin, as well as the probabilities of ruin are obtained. Finally, some numerical illustrations from a specific example are also given.     

Testing homogeneity against trend based on rank in one-way layout

In this paper, we are concerned with testing homogeneity against trend. Parsons (1979) considered the exact distribution of the test statistic based on the Wilcoxon type scores. We extend his result to the case of the general scores. Then we give a table of significance probabilities for the Fisher-Yates normal scores. We also study the asymptotic distribution of the test statis-tic based on the general scores under the null hypothesis, and the asymptotic relative efficiency against Bartholomew's likelihood ratio test assuming normality     

Ruin Probabilities of a Dual Markov-Modulated Risk Model

This article is devoted to studying a dual Markov-modulated risk model, which can properly represent, to some extent, surplus processes of companies that pay costs continuously and have occasional gains. We consider both the finite and infnite horizon ruin probabilities under this dual model. Upper and lower bounds of Lundberg type are derived for these ruin probabilities. We also obtain a time-dependent version of Lundberg type inequalities.     

Asymptotic Behavior of Random Time Ruin Probability Under Heavy-Tailed Claim Sizes and Dependence Structure

This article deals with the renewal risk model, in which there exists some asymptotic dependence relation between claim sizes and the inter-arrival times, and claim sizes are subexponential. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results.     

Subexponential Distributions — Large Deviations with Applications to Insurance and Queueing Models

This paper presents a fine large‐deviations theory for heavy‐tailed distributions whose tails are heavier than exp(?√t and have finite second moment. Asymptotics for first passage times are derived. The results are applied to estimate the finite time ruin probabilities in insurance as well as the busy period in a GI/G/1 queueing model.     

Ordres stochastiques et efficacité asymptotique des L-estimateurs

The efficiency of an estimator depends heavily on the tails of the distribution of the observations. Several partial orders have been defined to compare probability distributions according to their tails. In this paper we show that the asymptotic relative efficiency of two L-estimators with monotone weight functions is isotonic with respect to the partial orders defined by van Zwet (1964) and Lawrence (1975). We also give results concerning trimmed means.     

Asymptotics for the Finite Time Ruin Probability in the Renewal Model with Consistent Variation

Abstract

This paper investigates the finite time ruin probability in the renewal risk model. Under some mild assumptions on the tail probabilities of the claim size and of the inter-occurrence time, a simple asymptotic relation is established as the initial surplus increases. In particular, this asymptotic relation is requested to hold uniformly for the horizon varying in a relevant infinite interval. The uniformity allows us to consider that the horizon flexibly varies as a function of the initial surplus, or to change the horizon into any nonnegative random variable as long as it is independent of the risk system.     

The ruin probabilities of a discrete time risk model with one-sided linear claim sizes and dependent risks

This article investigates the ruin probabilities of a discrete time risk model with dependent claim sizes and dependent relation between insurance risks and financial risks. The risk-free and risky investments of an insurer lead to stochastic discount factors {θ_n}_{n ? 1}. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed (i.i.d.) innovations {?_n}_{n ? 1}. The i.i.d. random pairs {(?_n, θ_n)}_{n ? 1} follow a common bivariate Sarmanov-dependent distribution. When the common distribution of the innovations is heavy tailed, we establish some asymptotic estimates for the ruin probabilities of this discrete time risk model.     

Asymptotic properties of numbers of observations in random regions determined by central order statistics

In this paper, we extend the concept of near order statistic observation by considering observations that fall into a random region determined by a given order statistic and a Borel set. We study asymptotic properties of numbers of such observations as the sample size tends to infinity and the order statistic is a central one. We show that then proportions of these numbers converge in probability to some population probabilities. We also prove that these numbers can be centered and normalized to yield normal limit law. First, we derive results for one order statistic; next we give extensions to the multivariate case of two or more order statistics.     

The Ruin Probability of a Discrete-Time Risk Model with a One-Sided Linear Claim Process

In this article, the ruin probability is examined in a discrete time risk model with a constant interest rate, in which the dependent claims are assumed to have a one-sided linear structure. An explicit asymptotic formula is obtained for the ruin probability. Generalized Lundberg inequalities for the ruin probability are derived by martingale and inductive approaches.