Classical Risk Process Research Articles

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 However, when the analytical solution cannot be determined since the Laplace transform is no longer applicable due to the presence of an improper integral that is not possible to solve analytically.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

Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process

We find the asymptotics of the value function maximizing the expected utility of discounted dividend payments of an insurance company whose reserves are modeled as a classical Cramér risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We also perform some numerical analysis.

Risk process with mixture of tempered stable inverse subordinators: Analysis and synthesis

Abstract We propose two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterize the risk processes by deriving the marginal distributions and establish the moments and covariance structure. We study the main characteristics of these models such as ruin probability and time to ruin and illustrate the results with Monte Carlo simulations. The data suggest that the ruin time can be approximated by the inverse gaussian distribution and its generalizations.

Approximating the classical risk process by stable Lévy motion

The classical Cramér–Lundberg risk process is commonly used to model the surplus of an insurer; it characterizes the claim arrival process and the claim size random variable Y through a compound Poisson process, along with a constant rate of premium income. When , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto with . In this paper, we generalize the well known diffusion approximation by assuming that Y lies in the domain of attraction of an α-stable random variable, for . Then, we construct a sequence of classical Cramér–Lundberg risk processes and show that the sequence converges to an α-stable Lévy motion in the Skorokhod -topology. We prove this convergence by proving the pointwise convergence of the corresponding Laplace exponents of our processes, which to our knowledge, is a new result. To apply this convergence result, we show the convergence of a sequence of Gerber–Shiu distributions of exponential Parisian ruin, and we show the convergence of a sequence of payoff functions for barrier dividend strategies. Both of these applications provide new proofs of the stated limits.

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.

On Approximations to the Ruin Probability for the Classical Risk Process

On Approximations to the Ruin Probability for the Classical Risk Process

Optimization of the Cramer Lundberg model based value function of reinsurance with random claims and new premium arrival

In general, an insurance company who experiences two opposing cash flows incoming cash premiums and outgoing claims that is also known as classical risk process that satisfies Cramér–Lundberg model. However the arrival of the new premium holders and there cash flow over a period of time was not considered in most works. In this model, we considered the arrival of new premiums with expectation of surplus process until ruin time with dynamic reinsurance strategy. For attaining this condition, we formulated a Value function which is bounded and satisfied by the Hamilton Jacobi Bellman (HJB) partial differential equation. We apply the policy iteration method to find the maximum the surplus level and corresponding dynamic reinsurance strategy under excess of loss, quota share and stop loss reinsurance problems.

Reducing the Possibility of Ruin by Maximizing the Survival Function for the Insurance Company’s Portfolio

In this paper, the intention was to reduce the possibility of ruin in the insurance company by maximizing its survival function. This paper uses a perturbed classical risk process as the basic model. The basic model was later compounded by refinancing and return on investment. The Hamilton–Jacobi–Bellman equation and integro‐differential equation of Volterra type were obtained. The Volterra integro‐differential equation for the survival function of the insurance company was converted to a third‐order ordinary differential equation which was later converted into a system of first‐order ordinary differential equations. This system was then solved numerically using the fourth‐order Runge‐Kutta method. The results show that the survival function increases with the increase in the intensity of the counting process but decreases with an increase in the instantaneous rate of stock return and return volatility. This is due to the fact that the insurance company faces more risk. Thus, this paper suggests that in this situation, more investments should be made in risk‐free assets.

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.

Moments of discounted aggregate claims with dependence based on Spearman copula

This paper considers an extension to the classical compound Poisson risk process by introducing a dependence structure between the inter-claim time and the claim size. We adopt the Spearman copula for constructing the dependence with the purpose of covering a wide range of positive dependence and developing a convex approximation to some bivariate copulas. We study the Laplace transform of the moments of the discounted aggregate claims in this framework. Then we derive the explicit expressions for the first three moments of the discounted aggregate claims in the case of the exponential and Pareto-distributed claim sizes. Numerical examples are provided to measure the impact of dependence on the discounted aggregate claims and to illustrate the efficiency of the proposed approximation method.

Optimal Parisian-type dividend payments penalized by the number of claims for the classical and perturbed classical risk process

Optimal Parisian-type dividend payments penalized by the number of claims for the classical and perturbed classical risk process

On a dividend problem with random funding

We consider a modification of the dividend maximization problem from ruin theory. Based on a classical risk process we maximize the difference of expected cumulated discounted dividends and total expected discounted additional funding (subject to some proportional transaction costs). For modelling dividends we use the common approach whereas for the funding opportunity we use the jump times of another independent Poisson process at which we choose an appropriate funding height. In case of exponentially distributed claims we are able to determine an explicit solution to the problem and derive an optimal strategy whose nature heavily depends on the size of the transaction costs. Furthermore, the optimal strategy identifies unfavourable surplus positions prior to ruin at which refunding is highly recommended.

Diffusion Approximation of a Risk Model with Non-Stationary Hawkes Arrivals of Claims

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given.

Singular Initial-Value and Boundary-Value Problems for Integrodifferential Equations in Dynamical Insurance Models with Investments

We investigate two insurance mathematical models of the following behavior of an insurance company in the insurance market: the company invests a constant part of the capital in a risk asset (shares) and invests the remaining part in a risk-free asset (a bank account). Changing parameters (characteristics of shares), this strategy is reduced to the case where all the capital is invested in a risk asset. The first model is based on the classical Cramer–Lundberg risk process for the exponential distribution of values of insurance demands (claims). The second one is based on a modification of the classical risk process (the so-called stochastic premium risk process) where both demand values and insurance premium values are assumed to be exponentially distributed. For the infinite-time nonruin probability of an insurance company as a function of its initial capital, singular problems for linear second-order integrodifferential equations arise. These equations are defined on a semiinfinite interval and they have nonintegrable singularities at the origin and at infinity. The first model yields a singular initial-value problem for integrodifferential equations with a Volterra integral operator with constraints. The second one yields more complicated problem for integrodifferential equations with a non-Volterra integral operator with constraints and a nonlocal condition at the origin. We reduce the problems for integrodifferential equations to equivalent singular problems for ordinary differential equations, provide existence and uniqueness theorems for the solutions, describe their properties and long-time behavior, and provide asymptotic representation of solutions in neighborhoods of singular points. We propose efficient algorithms to find numerical solutions and provide the computational results and their economics interpretation.

Ruin probabilities for a perturbed risk model with stochastic premiums and constant interest force

In this paper, we consider a perturbed compound Poisson risk model with stochastic premiums and constant interest force. We obtain the upper bound and Lundberg-Cramér approximation for the infinite-time ruin probability, and consider the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. We show that the model in our paper has similar results to the classical risk process and some existing generalized models.

Gerber–Shiu functionals for classical risk processes perturbed by an [formula omitted]-stable motion

We study the Gerber–Shiu functional of the classical risk process perturbed by a spectrally negative α-stable motion. We provide representations of the scale functions of the process as an infinite series of convolutions of given functions. This, together with a result from Biffis and Kyprianou (2010), allows us to obtain a representation of the Gerber–Shiu functional as an infinite series of convolutions. Moreover, we calculate the Laplace transform and derive a defective renewal equation for the Gerber–Shiu functional, thus extending previous work of Furrer (1998) and of Tsai and Willmot (2002). We also obtain asymptotic expressions for the joint tail distribution of the severity of ruin and the surplus before ruin.

Dividend moments for two classes of risk processes with phase-type interclaim times

In this paper, we consider the distribution of discounted dividend pay- ments until ruin under a risk model with two independent classes of claims in which both of the two interclaim times have phase- type distributions and a constant dividend barrier. We obtain the integro-differential equations with boundary conditions for the moment- generating function of the sum of the discounted dividend payments un- til ruin. Explicit expressions for arbitrary moments of the discounted dividend payments are derived if the distribution of the two classes claim amount both belong to the rational family. Finally, numerical illustrations are presented to show how the results are applied.

Analysis of ruin measures for two classes of risk processes with stochastic income

In this paper, we consider the ruin measures for two classes of risk processes. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. Historically, it has been assumed that the premium size is a constant. In this contribution, the premium income arrival process is a Poisson process. In this framework, both the integro-differential equation and the Laplace transform for the expected discounted penalty function are established. Explicit expressions for the expected discounted penalty function are derived when the claim amount distributions belong to the rational family. Finally, Numerical examples are

The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds

In this paper, we consider a perturbed risk model with two independent classes of risks under multiple thresholds in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations with boundary conditions for the expected discounted penalty function. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.

Calculation of ruin probabilities for a dense class of heavy tailed distributions

In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.