Ruin Probability Research Articles

Asymptotic infinite-time ruin probabilities for a bidimensional time-dependence risk model with heavy-tailed claims

Asymptotic infinite-time ruin probabilities for a bidimensional time-dependence risk model with heavy-tailed claims

Simulation for Ruin Probabilities in Insurance with Sequence Markov Dependence Random Variables

The aim of this paper is to calculate ruin probabilities using Monte Carlo method for two models: i) classical risk model with claim amounts are homogeneous Markov chains; ii) generalized risk models with premiums amounts, claim amounts are homogeneous Markov chains. The sequence of random variables in the article is considered as a series of Markov dependent random variables. The main results of this paper are Lemma 3.1, Lemma 3.2 and Lemma 3.3, which have built mathematical formulas for the simulation of the probability of insurance models considered in this paper. From those lemmas, we build algorithms to simulate ruin probability for insurance models considered in this paper. From these algorithms, we build numerical results illustrating the problems posed in the paper. These results all show that when the initial capital increases, the ruin probability will decrease, and when the time increases, the ruin probability will increase. This result is consistent with the theory of the risk problem in insurance.

Uniform Asymptotics for Finite-time Ruin Probability in a Dependent Risk Model with General Stochastic Investment Return Process

Uniform Asymptotics for Finite-time Ruin Probability in a Dependent Risk Model with General Stochastic Investment Return Process

Ruin probability of a continuous-time model with dependence between insurance and financial risks caused by systematic factors

Ruin problems of continuous-time models with various dependence structures under risky investment are extensively studied in the literature. Those studies focus on dependence structures arising from insurance businesses themselves and pay few attentions to dependence between insurance risk and investment risk. This study considers a continuous-time Poisson risk model with insurance claims and investment return jumps shocked by common systematic risk factors. Systematic risk factors cause immediate jumps of investment returns but there are often random delays for insurers to settle incurred claims. The study further assumes that the arrival times of claims are delayed by a common random time to the ones of their corresponding investment return jumps. When claim sizes are heavy-tailed distributed, a uniform asymptotic estimate for ruin probabilities of the model is established and a numerical study is given to show the accuracy of the result. The result captures the effect of the dependence structures caused by systematic risk factors on the asymptotic behaviors of ruin probabilities and allows arbitrary dependence structures between claim sizes and their corresponding investment return jumps.

Three-Player Gambler’s Ruin Problem: Some Extensions

For calculating the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. However, the martingale approach is not appropriate to determine the ruin time of asymmetric game. In general, ruin probabilities in both cases, that is, symmetric and asymmetric games as well as the expected ruin time for the asymmetric games are still need to be calculated. The current work is also about three-player gambler’s ruin problem with some extensions. We provide expressions for the ruin time with or without ties when all the players have equal or unequal initial fortunes. Finally, the validity of the asymmetric game is also tested through a Monte Carlo simulation study.
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A three-player gambler's ruin problem: some extensions

For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.

A transient Cramér–Lundberg model with applications to credit risk

AbstractThis paper considers a variant of the classical Cramér–Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e. the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. As well as an exact analysis (in terms of transforms) of this ruin probability, an asymptotic analysis is performed, including an efficient importance-sampling-based simulation approach.The base model is extended in multiple dimensions: (i) we consider a model in which there may, in addition, be losses that do not correspond to defaults, (ii) then we analyze a model in which the individual obligors are coupled via a regime switching mechanism, (iii) then we extend the model so that between the losses the reserve process behaves as a Brownian motion rather than a deterministic drift, and (iv) we finally consider a set-up with multiple groups of statistically identical obligors.

Ruin probability of insurance companies in Nigeria using model with economic environment: Pre and post capitalization analysis

Ruin probability of insurance companies in Nigeria using model with economic environment: Pre and post capitalization analysis

Finite-time ruin probability of a perturbed risk model with dependent main and delayed claims

This paper considers a delayed claim risk model with stochastic return and Brownian perturbation in which each main claim may be accompanied with a delayed claim occurring after a stochastic period of time, and the price process of the investment portfolio is described as a geometric Lévy process. By means of the asymptotic results for randomly weighted sum of dependent subexponential random variables we obtain some asymptotics for finite-time ruin probability. A simulation study is also performed to check the accuracy of the obtained theoretical result via the crude Monte Carlo method.

How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.

De Vylder and Goovaerts' conjecture on homogeneous risk models with equalized claim amounts

De Vylder and Goovaerts (2000) made a conjecture on the comparison of the finite time ruin probability in a homogeneous risk model and the corresponding ruin probability in an associated model with equalized claim amounts. The conjecture, however, remains an open problem. In this paper, we provide a conjecture that is stronger than De Vylder and Goovaerts' conjecture and also provide sufficient conditions for the conjectures, which are more mathematically tractable than De Vylder and Goovaerts' conjecture and thus easier to work with. By using the sufficient conditions for the conjectures, we solve De Vylder and Goovaerts' conjecture when n=3, where n is the number of claims in the finite time.

Banach contraction principle, q-scale function and ultimate ruin probability under a Markov-modulated classical risk model

Suppose that risk reserves of an insurance company are governed by a Markov-modulated classical risk model with parameters modulated by a finite-state irreducible Markov chain. The main purpose of this paper is to calculate ultimate ruin probability that ruin time, the first time when risk reserve is negative, is finite. We apply Banach contraction principle, q-scale functions and Markov property to prove that ultimate ruin probability is the fixed point of a contraction mapping in terms of q-scale functions and that ultimate ruin probability can be calculated by constructing an iterative algorithm to approximate the fixed point. Unlike Gajek and Rudź (Insurance: Mathematics and Economics, 80 (2018), 45–53), our paper uses q-scale functions to obtain more explicit Lipschitz constant in Banach contraction principle in our case so that proofs of several Lemmas and theorems in their Appendix are unnecessary and some of their assumptions are confirmed in our case.

On asymptotic finite-time ruin probabilities of a new bidimensional risk model with constant interest force and dependent claims

In this article, consider a new continuous-time bidimensional renewal risk model with constant force of interest, in which every kind of business is assumed to pay two classes of claims called the first and second ones, respectively. Suppose that the first class of claim vectors form a sequence of independent and identically distributed random vectors following a general dependence structure which share a common renewal counting process, and the second class of claim vectors, independent of the first class of claim vectors, constitute another sequence of independent and identically distributed random vectors which arrive according to two different renewal counting process. For such a model, when the claims are assumed to be subexponential or belong to the intersection of long-tailed and dominatedly varying-tailed class, some asymptotic formulas on finite-time ruin probabilities are derived. The obtained results substantially extend some existing ones in the literature.

The finite element method: A high‐performing approach for computing the probability of ruin and solving other ruin‐related problems

The finite element method (FEM), despite being a very powerful tool for solving partial differential equations, has never been used to compute the probability of ruin, nor it has been applied to other kinds of ruin‐related problems, at least to the best of our knowledge. In this paper, in order to explore possible applications of the FEM in insurance mathematics, a high‐order FEM is proposed to solve problems related to ruin. Specifically, a FEM based on either quadratic, cubic, or quartic polynomials is developed to compute the probability of ruin and the (discounted) expected value of the assets of a firm in two previously published ruin models. Moreover, when dealing with the calculation of the probability of ruin, a suitable operator splitting technique is employed, such that the FEM variational formulation is used to approximate only spatial derivatives. Numerical experiments are presented showing that the FEM performs extraordinarily well. In fact, the solutions of the two problems considered can be obtained with an error (in the maximum norm) of order 10−5 or even smaller in a time smaller than a second. Finally, in accordance with already established theoretical results, the FEM reveals to be superconvergent at the boundaries of the finite elements.

The Reduction of Initial Reserves Using the Optimal Reinsurance Chains in Non-Life Insurance

The aim of the paper is to propose, and give an example of, a strategy for managing insurance risk in continuous time to protect a portfolio of non-life insurance contracts against unwelcome surplus fluctuations. The strategy combines the characteristics of the ruin probability and the values VaR and CVaR. It also proposes an approach for reducing the required initial reserves by means of capital injections when the surplus is tending towards negative values, which, if used, would protect a portfolio of insurance contracts against unwelcome fluctuations of that surplus. The proposed approach enables the insurer to analyse the surplus by developing a number of scenarios for the progress of the surplus for a given reinsurance protection over a particular time period. It allows one to observe the differences in the reduction of risk obtained with different types of reinsurance chains. In addition, one can compare the differences with the results obtained, using optimally chosen parameters for each type of proportional reinsurance making up the reinsurance chain.

Asymptotic behavior for finite-time ruin probabilities in a generalized bidimensional risk model with subexponential claims

Asymptotic behavior for finite-time ruin probabilities in a generalized bidimensional risk model with subexponential claims

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

This paper considers a nonstandard renewal risk model with constant force of interest, in which each main claim induces a delayed claim. Assume that the main claim sizes and the inter-arrival times obey a dependence structure, and so do the delayed claim sizes and the arrival times. Supposing that the distributions of the main and delayed claim sizes both have subexponential tails, asymptotic estimate for the finite-time ruin probability of this risk model is obtained, as the initial surplus tends to infinity.

Ruin problems for epidemic insurance

AbstractThe paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob.22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

Mixed fractional risk process

In this paper, we introduce a compound version of the mixed fractional Poisson process (MFPP). We obtain its mean, variance and the system of fractional differential equations that governs its state probabilities. By using the compound MFPP as claim process, we introduce a risk process, namely, the mixed fractional risk process (MFRP). It is shown that the MFRP is a martingale with respect to a suitable filtration. Its variance and covariance are obtained using which its long-range dependence property is established. Also, we show that its increments exhibit the short-range dependence property. Later, we consider a different fractional risk process based on the compound MFPP. We call it the mixed fractional risk process-II (MFRP-II) and it differs from the MFRP in terms of premium. For exponentially distributed claim sizes the probability density function of the ruin time of MFRP-II is obtained. An asymptotic behaviour of its finite time ruin probability is derived when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large.

On ruin probabilities with risky investments in a stock with stochastic volatility

On ruin probabilities with risky investments in a stock with stochastic volatility