Asymptotic ruin probability for a by-claim risk model with pTQAI claims and constant interest force

Abstract

In this paper, we consider a by-claim risk model with a constant rate of interest force, in which the main claims and the by-claims form a sequence of pTQAI nonnegative random variables and all their distributions belong to the dominatedly-varying heavy-tailed subclass. We obtain the asymptotically upper and lower bound formulas of the ultimate ruin probability for such a by-claim risk model. As its by-products, some interesting properties for pTQAI structure are also investigated. The results extend some existing ones in the literature.     

Uniform asymptotics for ruin probability of a two-dimensional dependent renewal risk model

ABSTRACT

In this paper we consider the tail behavior of a two-dimensional dependent renewal risk model with two dependent classes of insurance business, in which the claim sizes are governed by a common renewal counting process, and their inter-arrival times are dependent, identically distributed. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all time in an infinite interval. Moreover, we point out that the formula still holds uniformly for all time in an infinite interval for widely dependent random variables (r.v.s) under some conditions.     

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.     

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.     

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.     

Asymptotic ruin probabilities of a dependent renewal risk model based on entrance processes with constant interest rate

In this paper, the risk model with constant interest based on an entrance process is investigated. Under the assumptions that the entrance process is a renewal process and the claim sizes satisfy a certain dependency, which belong to the different heavy-tailed distribution classes, the finite-time and infinite-time asymptotic estimates of the risk model with constant interest force are obtained.     

Asymptotic ruin probabilities for a bidimensional risk model with heavy-tailed claims and non-stationary arrivals

Abstract

This article studies a bidimensional risk model, in which an insurer simultaneously confronts two kinds of claims sharing a common non-stationary arrival process. Assuming that the arrival process satisfies a large deviation principle and the claim-size distributions are heavy tailed, an asymptotic formula for the corresponding ruin probability of this bidimensional risk model is obtained.     

On the probability of ruin in a continuous risk model with two types of delayed claims

ABSTRACT

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouché theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.     

Sensitivity analysis on ruin probabilities with heavy-tailed claims

In this note, we consider the classical insurance risk model with heavy-tailed claim distributions. By using the Pollaczek–Khinchin Formula, we provide some sensitivity analysis on the ruin probability.     

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.     

On some probability distributions associated with random walks

The probability distribution of the total number of games to ruin in a gambler's ruin random walk with initial position n, the probability distribution of the total size of an epidemic starting with n cases and the probability distribution of the number of customers served during a busy period M/M/1 when the service starts with n waiting customers are identical. All these can be easily obtained by using Lagrangian expansions instead of long combinatorial methods. The binomial, trinomial, quadrinomial and polynomial random walks of a particle have been considered with an absorbing barrier at 0 when the particle starts its walks from a point n, and the pgfs. and the probability distributions of the total number of jumps (trials) before absorption at 0 have been obtained. The values for the mean and variance of such walks have also been given.     

A delayed dual risk model

In this article, we study a dual risk model with delays in the spirit of Dassios–Zhao. When a new innovation occurs, there is a delay before the innovation turns into a profit. We obtain large initial surplus asymptotics for the ruin probability and ruin time distributions. For some special cases, we get closed-form formulas. Numerical illustrations will also be provided.     

Estimate for the Finite-time Ruin Probability in the Discrete-time Risk Model with Insurance and Financial Risks

In this article, we consider a discrete-time risk model with insurance and financial risks. We derive some refinements of a general asymptotic formula for the finite-time ruin probability under the assumptions that the net losses follow a common distribution in the intersection between the subexponential class and the Gumbel maximum domain of attraction, and the stochastic discount factors of the risky asset have a common distribution with extended regular variation. The obtained asymptotic upper and lower bounds are transparent and computable.     

Uniform asymptotics for a non standard renewal risk model with CLWD heavy-tailed claims

Consider a non standard continuous-time renewal risk model with a constant force of interest, in which the claim sizes are assumed to be conditionally linearly wide dependent (CLWD) and belong to the intersection of dominatedly varying tailed and long tailed class, and inter-arrival times are assumed to be a sequence of independent and identically distributed random variables independent of the claim sizes. Under some technical conditions, we obtain an asymptotic formula for the tail probability of discounted aggregate claims, which holds locally uniform for all time horizon within a finite interval. When the claim sizes are further restricted to be consistently varying tailed, we show that this asymptotic formula is globally uniform for all time horizon within an infinite interval.     

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.     

Finite time ruin probability and structural density properties in the presence of dependence in insurance risk model

In the present paper, we consider the classical compound Poisson risk model with dependence between claim sizes and claim inter-arrival time. We attempt to analyze the approximation of finite time ruin probability. The finite time ruin probabilities are plotted for fixed threshold value associated to the claim inter-arrival time and also for fixed dependence parameter in Nelsen (2006) copula separately. Additionally, a general form for joint density of the interclaim times and claim sizes is considered. With respect to the classical Gerber-Shiu's (1998) function, first some structural density properties of dependent collective risk model is obtained. Then the ladder height probability density function of claim sizes is computed and the dependency structure investigated for Erlang interclaim time. As the application, some dependent models of the interclaim times and claim sizes are studied.     

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.