Lundberg Inequality Research Articles

Ruin Probability for Some Mixed Linear Exponential Family in Classical Risk Process

This article presents the probability of ruin for the classical risk process by including the density function of claims which satisfies a mixed linear exponential family. This can be defined as <img src=image/13428362_01.gif>, where <img src=image/13428362_02.gif>, <img src=image/13428362_03.gif>, <img src=image/13428362_20.gif> is a positive integer with <img src=image/13428362_04.gif> with <img src=image/13428362_05.gif>, <img src=image/13428362_06.gif>, <img src=image/13428362_07.gif>, and <img src=image/13428362_08.gif> is the canonical parameter. The main results show that the ordinary differential equation for the probability of ruin in the general case by using chain rule and mathematical induction technique is given in Theorem 2.2, the ordinary differential equation for some mixed linear exponential family when <img src=image/13428362_09.gif>, <img src=image/13428362_10.gif>, <img src=image/13428362_11.gif>, <img src=image/13428362_12.gif>, <img src=image/13428362_13.gif>, <img src=image/13428362_14.gif> is demonstrated in Theorem 2.3, and an explicit solution for the probability of ruin when the mixed linear exponential family satisfies the conditions which are <img src=image/13428362_15.gif>, <img src=image/13428362_16.gif>, <img src=image/13428362_10.gif> with <img src=image/13428362_17.gif>, and <img src=image/13428362_13.gif> is indicated in Theorem 2.4. Finally, we use MATLAB to generate the numerical simulations for the probability of ruin in the risk process that the number of claims is a Poisson process and the density function of claims satisfies a mixed linear exponential family and a gamma distribution under the conditions of Theorem 2.4 with the parameters <img src=image/13428362_18.gif>=1 and <img src=image/13428362_19.gif>=0.2. The numerical results reveal that the relative frequency of the ruin and the ruin probability also satisfy the Lundberg inequality which is the necessary condition for the ruin probability. In addition, the absolute values of its differences are small in order to confirm that the main results are correct.

Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance

This paper investigates optimal investment and reinsurance policies for an insurance company under a correlated risk model with common Poisson shocks. The goal of the insurance company is to minimize the ultimate ruin probability. By the dynamic programming principle, the Hamilton–Jacobi–Bellman (HJB for short) equation associated with this control problem is obtained. Since there is no explicit solution to the HJB equation, this paper alternates to find the minimal exponential upper bound of the ruin probability. The exponential upper bound of ruin probability is also called Lundberg inequality. Minimizing Lundberg inequality is equal to finding the maximal Lundberg coefficient. It turns out that the optimal investment and reinsurance polices are constant policies. Some numerical examples are given to illustrate the impact of the dependent structure and the investment chance on the upper bound.

Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplace transformation of the time when the surplus reaches a given level for the first time is discussed, and the expectation and its variance are obtained. Finally, we give the numerical examples.

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.

Cox risk model with correlated classes of business

A correlated aggregate claim model with m dependent classes of insurance business is constructed, in which claim occurrences of m classes relate to Cox process and these claim processes are correlated. This Cox risk model with correlated classes of business is first transformed to the Cox model of n independent risk processes. Then the generalized Lundberg exponent and Lundberg inequality are obtained by the martingale approach. Finally, the explicit expression of Lundberg exponent under a special condition is derived.

Survival probability and ruin probability of a risk model

In this paper, a new risk model is studied in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is p-thinning process. The integral representations of the survival probability are gotten. The explicit formula of the survival probability on the infinite interval is obtained in the special case—exponential distribution.The Lundberg inequality and the common formula of the ruin probability are gotten in terms of some techniques from martingale theory.

Optimal Proportional Reinsurance and Ruin Probability

In this paper, we consider optimal proportional reinsurance from an insurer's point of view to maximize the adjustment coefficient. We obtain the explicit expressions for the optimal results in the diffusion approximation case as well as in the jump-diffusion case. Further, we derive a sharper bound on the ruin probability. When the claim sizes are exponentially distributed, we can prove that the Lundberg's inequality also holds for the ruin probability in the diffusion approximation case. Some numerical examples are presented to show that the ruin probability in the diffusion approximation case sometimes underestimates the ruin probability.

Lundberg inequalities in a diffusion environment

In this paper we consider a Cox process driven by a Markov process. We Markovise the Cox process by adding the supplementary governing component and we find the extended generator A of such constructed process and its domain D( A) . As an example we discuss a risk process with a Coxian arrival process generated by a diffusion process. For this process we derive Lundberg inequality for the ruin probability.

Excess of Loss Reinsurance and the Probability of Ruin in Finite Horizon

AbstractThe upper bound provided by Lundberg's inequality can be improved for the probability of ruin in finite horizon, as Gerber (1979) has shown. This paper studies this upper bound as a function of the retention limit, for an excess of loss arrangement, and compares it with the probability of ruin.

Lundberg inequalities for a Cox model with a piecewise constant intensity

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li ) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li . In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

Lundberg inequalities for a Cox model with a piecewise constant intensity

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

Refinements and distributional generalizations of Lundberg's inequality

An upper bound is obtained for the tail of the total claims distribution in terms of a ‘new worse than used’ distribution. A simple bound exists when the claim size distribution is also new worse than used. The classical exponential bound is also refined when the claim size distribution is new worse than used in convex ordering, as well as for other classes of claim size distributions characterized by properties of the failure rate and the mean residual lifetime. Pareto bounds may be used when claim size moments are known. The compound geometric case and ruin probabilities are then considered.

Lundberg bounds on the tails of compound distributions

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.

Lundberg bounds on the tails of compound distributions

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.

An upper bound for the probability of ultimate ruin

Abstract In the classical compound Poisson risk model, Lundberg's inequality provides both an upper bound for, and an approximation to, the probability of ultimate ruin. The result can be applied only when the moment generating function of the individual claim amount distribution exists. In this paper we derive an upper bound for the probability of ultimate ruin when the moment generating function of the individual claim amount distribution does not exist.

Exponential inequalities for ruin probabilities in the Cox case

Abstract We consider an insurance model where the underlying point process is a Cox process. Using a martingale approach we obtain extensions of the classical Lundberg inequality.

Probability of ruin with variable premium rate

Abstract The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to...

Use of differential and integral inequalities to bound ruin and queuing probabilities

Summary Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. These equations contain terms which cause difficulty in obtaining analytical solutions. One partial way out of this difficulty is to omit the awkward terms, or modify them, to produce a more tractable equation. The price to be paid for this is that the equation becomes a differential or integro-differential inequality. The paper concentrates on applying the theory of such inequalities to obtain bounds on probability of ruin over finite and infinite times respectively. By these methods it is possible to 1. (1) sharpen Lundberg's inequality;2. (2) order (in some cases) probabilities of ruin over infinite time in accordance with the properties of different claim size distributions;3. (3) give an upper bound for finite-time probability of non-ruin in terms only of probabilities involving (a) zero initial reserves, and (b) infinite time. Finally, the application of these results to queuing theory is indicated.