Ruin Probability Research Articles

Asymptotics for a bidimensional delay-claim risk model with subexponential claims and arbitrary dependence between the generic inter-arrival time pair

This article considers a bidimensional delay-claim renewal risk model with a constant interest force, in which it is assumed that each main claim and its corresponding delayed claim from one business line satisfy a general dependence structure and the generic inter-arrival time pair from the two lines of main claims is arbitrarily dependent. In the presence of subexponential main and delayed claims, the corresponding asymptotic formulae for four types of finite-time ruin probabilities are established which extend some existing ones in the literature.

RUIN PROBABILITY IN THE CLASSICAL RISK PROCESS WITH WEIBULL CLAIMS DISTRIBUTION

In the classical risk process, ruin is the situation when the surplus falls below zero. Ruin probability is a tool used to predict bankruptcy in the insurance company. The ruin probability can be determined by solving the Integral-Differential equation that arises from the classical risk process. In this paper, we are interested in calculating the ruin probability when the claim distribution follows the Weibull distribution. Based on the Weibull parameter, the calculation is divided into two cases: when alpha equals 1 and when . The Laplace transform gives the analytical solution of the Integral-Differential equation. However, when the analytical solution cannot be determined since the Laplace transform is no longer applicable due to the presence of an improper integral that is not possible to solve analytically. Therefore, for the case alpha greater than 1, Euler’s method is applied to determine its numerical solution. The accuracy of the numerical solution is validated by comparing it with the analytical solution for the case Then, using the accuracy determined from the first case, we apply the Euler method to determine the numerical solution for the case . The numerical method gives good accuracy to the analytical solution with the order of calculated from until

Ordering Results of Aggregate Claim Amounts from Two Heterogeneous Portfolios

In this paper, we discuss the stochastic comparison of two classical surplus processes in an one-year insurance period. Under the Marshall-Olkin extended Weibull random aggregate claim amounts, we establish some sufficient conditions for the comparison of aggregate claim amounts in the sense of the usual stochastic order. Applications of our results to the Value-at-Risk and ruin probability are also given. The obtained results show that the heterogeneity of the risks in a given insurance portfolio tends to make the portfolio volatile, which in turn leads to requiring more capital. We also obtain some sufficient conditions for comparing aggregate non-random claim amounts with different occurrence frequency vectors in terms of increasing convex order.

Uniform asymptotics for finite-time ruin probabilities of a bidimensional compound risk model with stochastic returns

Consider a continuous-time bidimensional compound risk model with stochastic returns, where an insurance company operates two lines of business at the same time and is allowed to invest its wealth into financial assets. In this model, each accident may cause a random number of heavy-tailed claims and the claim sizes from the same line of business are upper tail asymptotically independent, while the accident arrival processes from different lines of business are arbitrarily dependent. Under some moment conditions on the accident arrival processes, some uniform asymptotic formulae for finite-time ruin probabilities are established.

Abel-Gontcharoff polynomials, parking trajectories and ruin probabilities

Abstract The central mathematical tool discussed is a non-standard family of polynomials, univariate and bivariate, called Abel-Goncharoff polynomials. First, we briefly summarize the main properties of this family of polynomials obtained in the previous work. Then, we extend the remarkable links existing between these polynomials and the parking functions which are a classic object in combinatorics and computer science. Finally, we use the polynomials to determine the non-ruin probabilities over a finite horizon for a bivariate risk process, in discrete and continuous time, assuming that claim amounts are dependent via a partial Schur-constancy property.

Inequalities on the ruin probability for light-tailed distributions with some restrictions

When there are some restrictions on the random variables of insurance risk model, it is impossible to calculate the exact value of ruin probabilities. For these cases, even finding a suitable approximation, is very important from a practical point of view. In the present paper, we consider the renewal insurance surplus model with light-tailed claim amount distributions and try to find some inequalities on the infinite time ruin probability depending on the amount of initial reserve using statistical and mathematical approaches if the assumption of net profit does not hold but there exist some other restrictions on the mathematical functions of random variables of model. The assertions depend on the amount of initial reserve, distribution of nonnegative claim occurrences times and successive claim amounts are obtained. Finally, to show the application and effectiveness of given theorems two examples are presented. Through these examples, the infinite time ruin probabilities are estimated using Monte Carlo simulation and give an intuitive way to understand the nature of ruin.

Cramér–Lundberg asymptotics for spectrally positive Markov additive processes

This paper studies the Cramér–Lundberg asymptotics of the ruin probability for a model in which the reserve level process is described by a spectrally-positive light-tailed Markov additive process. By applying a change-of-measure technique in combination with elements from Wiener-Hopf theory, the exact asymptotics of the ruin probability are expressed in terms of the model primitives. In addition a simulation algorithm of generalized Siegmund type is presented, under which the returned estimate of the ruin probability has bounded relative error. Numerical experiments show that, compared to direct estimation, this algorithm greatly reduces the number of runs required to achieve an estimate with a given accuracy. The experiments also reveal that our asymptotic results provide a good approximation of the ruin probability even for relatively small initial surplus levels.

Parisian ruin with power-asymmetric variance near the optimal point with application to many-inputs proportional reinsurance

. This article investigates the Parisian ruin probability for a class of Gaussian processes with power-asymmetric behavior of the variance near the unique optimal point. We derive the exact asymptotics as the initial capital tends to infinity and extend the previous result[8] to the case when the length of the Parisian interval is on the Pickands scale. As a primary application, we extend the recent result[13] on the many inputs proportional reinsurance fractional Brownian motion risk model to the Parisian ruin.

The Sine Modified Power-Generated Family of Distributions with Application to Practical Data and Ruin Probability

The Sine Modified Power-Generated Family of Distributions with Application to Practical Data and Ruin Probability

Estimating a VaR-type ruin measure by Laguerre series expansion in classical compound Poisson risk model

Value at Risk (VaR) has been extensively applied within the banking and financial sectors. Regulators use it to reflect riskiness and determine capital requirements. In this paper, we propose an efficient method for estimating a VaR-type ruin measure under the classical risk model. We first use Laguerre series expansion to approximate the ruin probability and utilize a tabularization method to approximate the VaR-type ruin measure. A practical scenario where the distribution of individual claim size and the Poisson intensity are unknown has been considered. Based on observed data providing claim information, we illustrate the estimation of the VaR ruin measure. Moreover, the approximation convergence rates and statistical errors are studied, respectively. Various numerical examples are provided to demonstrate the performance of the proposed method.

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the sizes of the incurred random claims are on average less than the premiums gained between the successive interoccurrence times. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims’ interoccurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims are on average equal to the premiums gained between the successive interoccurrence times. We study the discrete-time risk model with N ∈ N periodically occurring independent distributions, the classical risk model, also known as the Cramér–Lundberg risk process, and the more general Sparre Andersen model.

On a time-changed variant of the generalized counting process

AbstractIn this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

Poissonian occupation times of refracted Lévy processes with applications

Inspired by the work of Lkabous (2021), we consider Poissonian occupation times below level 0 of a refracted Lévy process where its premium rate is adaptive. In this model, occupation time is accumulated once the surplus process is observed to be negative at Poisson arrival times. Our analysis depends on various exit identities for refracted Lévy processes observed at Poisson arrival times. As an application of Poissonian occupation times, we derive an explicit expression for the probability of Parisian ruin with Erlang(2, λ) implementation delays. The resulting main quantities are in terms of scale functions.

Research on a Model for an Empirical Analysis of Inherent Defect Insurance Based on Ruin Theory

The system of inherent defect insurance is an important measure to serve the real economy through financial means and improve the quality of construction projects, which is the future development direction of China’s construction industry. However, the related research is not perfect due to the short implementation of the insurance. This could bring risks to the promotion of insurance for companies. Insurance ruin theory is an important method in risk management theory, so adopting it to manage the risk inherent defect insurance from the perspective of insurance companies is vital. The research starts with the classic insurance ruin theory and determines the coefficient of premium collection from the perspective of claim settlement distribution expectations. Furthermore, the approximate distribution of claim settlement is deduced, and a comprehensive risk assessment model is constructed. Finally, based on the data of insurance actuarial practice in Shanghai, both the ruin probability of inherent defect insurance in each insuring term and its average required initial reserve are calculated, which provides the analyses on the risks and main subitems of inherent defect insurance as well as relevant suggestions. Finally, the sensitivity analysis is used to further analyse the risk of different insurance stages of IDI, and the relevant measures are proposed. The research can provide theoretical assistances for insurance companies to carry out effective risk management and provide model tools to make scientific decisions.

Quantum Encoding and Analysis on Continuous Time Stochastic Process with Financial Applications

Modeling stochastic phenomena in continuous time is an essential yet challenging problem. Analytic solutions are often unavailable, and numerical methods can be prohibitively time-consuming and computationally expensive. To address this issue, we propose an algorithmic framework tailored for quantum continuous time stochastic processes. This framework consists of two key procedures: data preparation and information extraction. The data preparation procedure is specifically designed to encode and compress information, resulting in a significant reduction in both space and time complexities. This reduction is exponential with respect to a crucial feature parameter of the stochastic process. Additionally, it can serve as a submodule for other quantum algorithms, mitigating the common data input bottleneck. The information extraction procedure is designed to decode and process compressed information with quadratic acceleration, extending the quantum-enhanced Monte Carlo method. The framework demonstrates versatility and flexibility, finding applications in statistics, physics, time series analysis and finance. Illustrative examples include option pricing in the Merton Jump Diffusion Model and ruin probability computing in the Collective Risk Model, showcasing the framework’s ability to capture extreme market events and incorporate history-dependent information. Overall, this quantum algorithmic framework provides a powerful tool for accurate analysis and enhanced understanding of stochastic phenomena.

Ruin probabilities for a Sparre Andersen model with investments: the case of annuity payments

Ruin probabilities for a Sparre Andersen model with investments: the case of annuity payments

Ruin in a continuous-time risk model with arbitrarily dependent insurance and financial risks triggered by systematic factors

This paper is devoted to asymptotic analysis for a continuous-time risk model with the insurance surplus process and the log-price process of the investment driven by two dependent jump-diffusion processes. We take into account arbitrary dependence between the insurance claims and their corresponding investment return jumps caused by a sequence of systematic factors, whose arrival times constitute a renewal counting process. Under the framework of regular variation, we obtain a simple and unified asymptotic formula for the finite-time ruin probability as the initial wealth becomes large. It turns out that, in the weakly dependent case, the tails of the claims determine the exact decay rate of the finite-time ruin probability while the investment return jumps only contribute to the coefficient of the asymptotic formula; however, in the strongly dependent case, they both produce essential impacts on the finite-time ruin probability which is under-estimated in the weakly dependent case.

Some Stochastic Orders over an Interval with Applications

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds.

Ruin Theory for User Association and Energy Optimization in Multi-Access Edge Computing

In this correspondence, a novel framework is proposed for analyzing data offloading in a multi-access edge computing system. Specifically, a two-phase algorithm, is proposed, including two key phases: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1) user association phase</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2) task offloading phase</i> . In the first phase, a ruin theory-based approach is developed to obtain the users association considering the users' transmission reliability and resource utilization efficiency. Meanwhile, in the second phase, an optimization-based algorithm is used to optimize the data offloading process. In particular, ruin theory is used to manage the user association phase, and a ruin probability-based preference profile is considered to control the priority of proposing users. Here, ruin probability is derived by the surplus buffer space of each edge node at each time slot. Giving the association results, an optimization problem is formulated to optimize the amount of offloaded data aiming at minimizing the energy consumption of users. Simulation results show that the developed solutions guarantee system reliability, association efficiency under a tolerable value of surplus buffer size, and minimize the total energy consumption of all users.

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.