Ruin Probability Research Articles

Sojourns of locally self-similar Gaussian processes

The cumulative Parisian ruin probability over a finite time interval [ 0 , T ] for a Gaussian risk process R ( t ) = u + c ( t ) − X ( t ) , t ≥ 0 , is the probability that the total time the process R ( t ) spends below zero during this interval exceeds a threshold L ≥ 0 . In this contribution we derive exact asymptotic approximations of the cumulative Parisian ruin probability for a general class of Gaussian processes introduced in D ȩ bicki and Tabiś [Pickands-Piterbarg constants for self-similar Gaussian processes, Probab. Math. Statist. 40 (2020), pp. 297–315. MR 4206416] assuming that X is locally self-similar. We illustrate our findings with several examples. As a byproduct we show that Berman's constants can be defined alternatively by a self-similar Gaussian process which could be quite different to the fractional Brownian motion.

Ruin probability approximation for bidimensional Brownian risk model with tax

Let B(t)=(B1(t),B2(t)), t≥0 be a two-dimensional Brownian motion with independent components and define the γ-reflected process X(t)=(X1(t),X2(t))=B1(t)−c1t−γ1infs1∈[0,t](B1(s1)−c1s1),B2(t)−c2t−γ2infs2∈[0,t](B2(s2)−c2s2),with given finite constants c1,c2∈R and γ1,γ2∈[0,2). The goal of this paper is to derive the asymptotics of the ruin probability P∃t∈[0,T]:X1(t)>u,X2(t)>auas u→∞ for T>0.

Computational Methods of Ruin Probability: Actuarial Comparison of De-Vylder and Tijim’s Models

The underwriting operation of insurance firms is to assume the risk of the insured in return of premium received. In order to shield itself against extreme losses and avoid the risk of insolvency, it becomes necessary to examine how the portfolio is expected to perform over a long-time horizon. The surplus process is connected with the excess of the premium received over claims outgo of the insurer’s portfolio to enable it predict the level at which the insurer could survive. When the surplus approaches a defined lower limit irrespective of the initial reserve, then the insurer is ruined. The probability of ruin apparently defines the volatility embedded in underwriting process as a useful tool of risk measurement in long range planning of premium rate. This paper is anchored on the following objectives: solve the adjustment co-efficient using the moment generating function, compute the De-Vylder’s ruin, compute the Tijim’s ruin approximation and then compare the two. Although it was verified that as the initial capital increases, the probability of ruin decreases under the two models, computational evidence from our results in tables 2 and 3 shows that the Tijim’s approximation to ruin probability is higher than the De-Vylder’s approximation at the same level of initial capital and safety loading. The implication is that the De-Vylder’s approximation to ruin probabilities is an improvement over Tijim’s ruin model and hence De-Vylder’s approximation is recommended for the insurer’s ruin assessment

Advantages of Accounting for Stochasticity in the Premium Process

In this paper, we study a risk model with stochastic premium income and its impact on solvency risk management. It is assumed that both the premium arrival process and the claim arrival process are modelled by homogeneous Poisson processes, and that the premium amounts are modelled by independent and identically distributed random variables. While this model has been studied in the existing literature and certain explicit results are known under more restrictive assumptions, these results are relatively difficult to apply in practice. In this paper, we investigate the factors that differentiate this model and the classical risk model. After reviewing various known results of this model, we derive a simulation approach for obtaining the probability of ultimate ruin based on importance sampling, which does not require specific distributions for the premium and the claim. We demonstrate this approach first with examples where the distribution of the sampling random variable can be identified. We then provide additional examples where we use the fast Fourier transform to obtain an approximation of the sampling random variable. The simulated results are compared with the known results for the probability of ruin. Using the simulation approach, we apply this model to a real-life auto-insurance data set. Differences with the classical model are then discussed. Finally, we comment on the suitability and impact of using this model in the context of solvency risk management.

Asymptotics for Finite-Time Ruin Probabilities of a Dependent Bidimensional Risk Model with Stochastic Return and Subexponential Claims

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ψand(x1,x2;T) and ψsim(x1,x2;T) have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ψor(x1,x2;T) is given.

A Threshold Estimator for Ruin Probability Using the Fourier-Cosine Method in the Wiener–Poisson Risk Model

In this paper, we propose a nonparametric estimator of ruin probability in the Wiener–Poisson risk model based on high-frequency data. The estimator is constructed via the Fourier-cosine method and the threshold technique, and the convergence rate is also studied for a large sample size. Finally, we verify the effectiveness of our estimator through some simulation studies.

Robust estimator of the ruin probability in infinite time for heavy-tailed distributions

The probability of ruin of an insurance company is one of the main risk measures considered in risk theory, and the problems of its calculation and approximation have attracted much attention. Statistical estimations have been developed on the ruin probability in infinite time for insurance loses from heavy-tailed distributions. However, these estimation suffer heavily from under-coverage or have a robustness problem. We therefore need another method for estimating the probability of ruin in infinite time for heavy-tailed losses. In this paper, we introduce a robust estimator of the infinite-time probability of ruin for such distributions. Our methodology is based on extreme value theory, which offers adequate statistical results for such distributions. Our approach is based on a sensitive distribution known as the t-Hill estimator (t-score or score moment estimation) for the index of any tail distribution and introduced in Fabiáan and Stehlík. We establish their asymptotic normality, and through a simulation study, illustrate their behaviour in terms of absolute bias and mean squared error. The simulation results show that our estimators perform well and that they are fairly robust to outliers.

Infinite-Horizon Probability of Ruin for a Variable-Memory Counting Process (Hawkes Process)

The reserve R(t) of an insurance company denotes the accumulated capital up to time ttt throughout its operational span. While this reserve can be modeled using stochastic processes, the endeavor remains notably intricate. The mathematical complexity presents a considerable obstacle for researchers within this domain. Prior research has predominantly focused on reserve models constructed from Markovian processes. In this manuscript, our attention shifts to risk models derived from non-Markovian processes, particularly those with claim arrivals governed by Hawkes processes. Before delving into the core subject matter, we provide an extensive review of the literature on counting processes with variable memory. This includes a thorough exploration of Hawkes processes, Gerber-Shiu functions, integro-differential equations, and Laplace transforms. These elements are crucial for computing the probability of ruin over an infinite time horizon, especially in scenarios where interdependence exists between inter-claim times. Additionally, we present the requisite mathematical tools essential for comprehending the contents of this article, specifically tailored for individuals with a foundational understanding of actuarial science. Furthermore, we have successfully determined the probability of ruin, marking a significant milestone in our investigation.

Robust Risk Control with Reinsurance and CAT Bonds

This research studies robust risk control policies for an ambiguity-averse insurer. The insurer can manage its risk exposure through the purchase of reinsurance and the issuance of parametric catastrophe (CAT) bonds, which are linked to an exogenous trigger index. The insurer worries about the veracity of the model and seeks to develop a robust strategy to minimize the discounted ruin probability. We allow the insurer to exhibit different levels of ambiguity aversion toward its own claims and the trigger index. By employing a robust control approach, we analytically derive the optimal risk control policies that provide the insurer with guidelines on how to effectively manage its risk in an ambiguous environment. We present numerical examples that showcase scenarios in which CAT bonds can be utilized as an effective risk mitigation tool. Furthermore, we assess the potential welfare losses that could arise if the insurer fails to account for model uncertainty or lacks the ability to issue CAT bonds.

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

Finite-time ruin probability of a risk model with perturbation and subexponential main claims and by-claims

The paper considers a nonstandard risk model with stochastic return and perturbation, in which the price process of the investment portfolio is described as a geometric Lévy process and each main claim may induce a delayed by-claim. When the main claims and by-claims have subexponential distributions, we obtain some asymptotic estimations of the finite-time ruin probability.

Asymptotic estimates for ruin probabilities in a bidimensional delay-claim risk model with subexponential claims

This article studies the asymptotic behaviors for some common ruin probabilities of a bidimensional delay-claim risk model, in which each claim follows a subexponential distribution and each claim-size inherits some dependence structure. Meanwhile, the asymptotic behavior for sum ruin probability of the bidimensional delay-claim risk model is investigated when the two components of each claim inter-arrival time vector are arbitrarily dependent.

Asymptotics for ruin probabilities of a bidimensional risk model with a random number of delayed claims

This article considers a non-standard bidimensional risk model with constant force of interest, in which every main claim may cause a random number of delayed claims in an uncertain period of time. Specifically, if both the main claims and delayed claims are subexponential and follow some dependence structure, we drive precise asymptotic formulae for finite-time ruin probabilities, which generalize the results of Lu and Yuan (2022) and Li (2023b) to some extent.

Reduced-bias estimator of the ruin probability in infinite time for heavy-tailed distributions with index in the upper half of the unit interval

Reduced-bias estimator of the ruin probability in infinite time for heavy-tailed distributions with index in the upper half of the unit interval

Analyzing Sequential Betting with a Kelly-Inspired Convective-Diffusion Equation.

The purpose of this article is to analyze a sequence of independent bets by modeling it with a convective-diffusion equation (CDE). The approach follows the derivation of the Kelly Criterion (i.e., with a binomial distribution for the numbers of wins and losses in a sequence of bets) and reframes it as a CDE in the limit of many bets. The use of the CDE clarifies the role of steady growth (characterized by a velocity U) and random fluctuations (characterized by a diffusion coefficient D) to predict a probability distribution for the remaining bankroll as a function of time. Whereas the Kelly Criterion selects the investment fraction that maximizes the median bankroll (0.50 quantile), we show that the CDE formulation can readily find an optimum betting fraction f for any quantile. We also consider the effects of "ruin" using an absorbing boundary condition, which describes the termination of the betting sequence when the bankroll becomes too small. We show that the probability of ruin can be expressed by a dimensionless Péclet number characterizing the relative rates of convection and diffusion. Finally, the fractional Kelly heuristic is analyzed to show how it impacts returns and ruin. The reframing of the Kelly approach with the CDE opens new possibilities to use known results from the chemico-physical literature to address sequential betting problems.

Ruin probability for heavy-tailed and dependent losses under reinsurance strategies

The frequency and severity of extreme events have increased in recent years in many areas. In the context of risk management for insurance companies, reinsurance provides a safe solution as it offers coverage for large claims. This paper investigates the impact of dependent extreme losses on ruin probabilities under four types of reinsurance: excess of loss, quota share, largest claims, and ecomor. To achieve this, we use the dynamic GARCH-EVT-Copula combined model to fit the specific features of claim data and provide more accurate estimates compared to classical models. We derive the surplus processes and asymptotic ruin probabilities under the Cramér–Lundberg risk process. Using a numerical example with real-life data, we illustrate the effects of dependence and the behavior of reinsurance strategies for both insurers and reinsurers. This comparison includes risk premiums, surplus processes, risk measures, and ruin probabilities. The findings show that the GARCH-EVT-Copula model mitigates the over- and under-estimation of risk associated with extremes and lowers the ruin probability for heavy-tailed distributions.

Probabilistic approach to risk processes with level-dependent premium rate

We study risk processes with level dependent premium rate. Assuming that the premium rate converges, as the risk reserve increases, to the critical value in the net-profit condition, we obtain upper and lower bounds for the ruin probability; our proving technique is purely probabilistic and based on the analysis of Markov chains with asymptotically zero drift.We show that such risk processes give rise to heavy-tailed ruin probabilities whatever the distribution of the claim size, even if it is a bounded random variable. So, the risk processes with near critical premium rate provide an important example of a stochastic model where light-tailed input produces heavy-tailed output.

Empirical risk analysis of mining a Proof-of-Work blockchain

AbstractThe process of mining blocks on a blockchain utilizing a Proof-of-Work consensus mechanism carries inherent risks, particularly when the operational expenses associated with mining exceed the rewards earned. Building on previous findings on mining in pools, this paper delves into the question of whether the theoretical formulas for the ruin probability and the expected value of future surplus obtained under particular model assumptions are indeed validated empirically. In particular, we include the presence of transactions fees in the block rewards in our analysis. We also provide algorithms to fit the involved generalized hyperexponential distributions to actual data. Moreover, we perform a sensitivity analysis for different factors of interest, and we quantify the relevance of incorporating temporal dependence and transaction fees in the model.

Ruin Probabilities with Investments in Random Environment: Smoothness

Ruin Probabilities with Investments in Random Environment: Smoothness

Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management.