Absolute Ruin Probability Research Articles

A Note on Gerber–Shiu Function with Delayed Claim Reporting under Constant Force of Interest

In this paper, we analyze the Gerber–Shiu discounted penalty function for a constant interest rate in delayed claim reporting times. Using the Poisson claim arrival scenario, we derive the differential equation of the Laplace transform of the generalized Gerber–Shiu function and show that the differential equation can be transformed to a Volterra equation of the second kind with a degenerated kernel. In the case of an exponential claim distribution, a closed-expression for the Gerber–Shiu function is obtained via sequence expansion. This result allows us to calculate the absolute (relative) ruin probability. Additionally, we discuss a method of solving the Volterra equation numerically and provide an illustration of the ruin’s probability to support the finding.

Minimizing the probability of absolute ruin under the mean‐variance premium principle

AbstractIn this article, we assume that the insurer can purchase per‐loss reinsurance and invest its surplus in a financial market consisting of a risk‐free asset and a risky asset. It is also assumed that the investment amount of the risky asset is capped at a fixed level and that short‐selling is prohibited. Our objective is to minimize the probability of absolute ruin, and the reinsurance premium is computed according to the mean‐variance premium principle, that is, a combination of the expected‐value and variance premium principles. By solving the corresponding Hamilton–Jacobi–Bellman equation, we derive explicit expressions for the S‐shaped minimum absolute ruin function and its associated optimal reinsurance‐investment strategy. We further study the same optimization problem for a slightly modified version of absolute ruin, and the corresponding optimal results are obtained as well. To gain insights into the optimal problems, we investigate the reason for the S‐shaped value function and discover that the constraint on investment control can result in the kink of minimum absolute ruin function. Finally, some properties and numerical examples are presented to show the impact of model parameters on the optimal results.

Minimizing the Probability of Absolute Ruin Under Ambiguity Aversion

In this paper, we consider an optimal robust reinsurance problem in a diffusion model for an ambiguity-averse insurer, who worries about ambiguity and aims to minimize the robust value involving the probability of absolute ruin and a penalization of model ambiguity. It is assumed that the insurer is allowed to purchase per-claim reinsurance to transfer its risk exposure, and that the reinsurance premium is computed according to the mean-variance premium principle which is a combination of the expected-value and variance premium principles. The optimal reinsurance strategy and the associated value function are derived explicitly by applying stochastic dynamic programming and by solving the corresponding boundary-value problem. We prove that there exists a unique point of inflection which relies on the penalty parameter greatly such that the robust value function is strictly concave up to the unique point of inflection and is strictly convex afterwards. It is also interesting to observe that the expression of the optimal robust reinsurance strategy is independent of the penalty parameter and coincides with the one in the benchmark case without ambiguity. Finally, some numerical examples are presented to illustrate the effect of ambiguity aversion on our optimal results.

Closed-Form Absolute Ruin Problems of the Risk Models with State-Dependent Switched Claims

This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.

Infinite-time absolute ruin in dependent renewal risk models with constant force of interest

ABSTRACTConsider a renewal risk model with a constant premium and a constant force of interest rate, where the claim sizes and inter-arrival times follow certain dependence structures via some restriction on their copula function. Under the assumption that the distribution of the claim-size belongs to the intersection of the class and the class , or a larger intersection class of O-subexponential distribution, class and , the infinite-time absolute ruin probabilities are derived.

Minimization of absolute ruin probability under negative correlation assumption

In this paper we consider the problem of minimizing the absolute ruin probability of an insurance company. The managers of the company control investment amount and risk exposure to minimize the absolute ruin probability. A negative correlation between insurer’s liabilities and capital gains in financial market is introduced. Under this negative correlation assumption, the explicit forms of the solutions and optimal strategies to this problem for all different parameters are derived. We find that the solutions of this problem are S-shaped and the optimal strategies fail to be monotonic or continuous.

Optimal Dynamic Risk Control for Insurers with State-Dependent Income

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Optimal Dynamic Risk Control for Insurers with State-Dependent Income

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Optimal Dynamic Risk Control for Insurers with State-Dependent Income

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

The Perturbed Dual Risk Model with Constant Interest and a Threshold Dividend Strategy

We consider the perturbed dual risk model with constant interest and a threshold dividend strategy. Firstly, we investigate the moment-generation function of the present value of total dividends until ruin. Integrodifferential equations with certain boundary conditions are derived for the present value of total dividends. Furthermore, using techniques of sinc numerical methods, we obtain the approximation results to the expected present value of total dividends. Finally, numerical examples are presented to show the impact of interest on the expected present value of total dividends and the absolute ruin probability.

Asymptotics for the in nite-time absolute ruin probabilities in time-dependent renewal risk models

In this paper, we consider the in nite-time absolute ruin probabilities in two types of time-dependent renewal risk models with a constant premium rate and a constant interest rate. In the two models, we both as- sume that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys some dependence structure. We derive an asymptotic formula and an upper bound for the in nite-time absolute ruin probabilities in the two dependent cases, respec- tively, with the claim-size distribution belonging to the intersection of the class J ( γ ), γ > 0, and the class R -∞.

Absolute ruin in the compound Poisson model with credit and debit interests and liquid reserves

In this paper, we study the absolute ruin probability in the compound Poisson model with credit and debit interests and liquid reserves. At first, we derive a system of integro‐differential equations with certain boundary conditions for the Gerber–Shiu function. Then, applying these results, we obtain asymptotical formula of the absolute ruin probability for subexponentially claims. Furthermore, when the claims are exponentially distributed, we obtain the explicit expressions for the Gerber–Shiu function and the exact solution for the absolute ruin probability. Finally, we discuss the absolute ruin probability by using the Gerber–Shiu function when debit interest is varying. In the case of exponential individual claim, we give the explicit expressions for the Gerber–Shiu function. Copyright © 2012 John Wiley & Sons, Ltd.

Asymptotics and uniform asymptotics for finite-time and infinite-time absolute ruin probabilities in a dependent compound renewal risk model

In this paper, we consider a dependent compound renewal risk model with constant premium rate and interest rate, where the individual claim sizes are widely orthant dependent and the claim number has a distribution belonging to the intersection among the maximum domain of attraction of the Gumbel distribution, the subexponential class and the rapidly-varying class. In such a dependent compound renewal (or Poisson) risk model, we obtain the asymptotics and uniform asymptotics for the finite-time and infinite-time absolute ruin probabilities. To this end, we investigate the tail behavior of the random sum with some widely orthant dependent summands and the random number in the maximum domain of attraction of the Gumbel distribution.

Asymptotic behavior of random time absolute ruin probability with [formula omitted] tailed and conditionally independent claim sizes

Consider the probability of random time absolute ruin in the renewal risk model with constant premium rate and constant force of interest. We assume that claim sizes Xi,i=1,2,…, are conditionally independent on some sigma algebra and that the common distribution belongs to class D∩L. We obtain the asymptotic formula for the subclass of subexponential distributions.

On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest

We consider a compound Poisson surplus process perturbed by diffusion with debit interest. When the surplus is below zero or the company is on deficit, the company is allowed to borrow money at a debit interest rate to continue its business as long as its debt is at a reasonable level. When the surplus of a company is below a certain critical level, the company is no longer profitable, we say that absolute ruin occurs at this situation. In this risk model, absolute ruin may be caused by a claim or by oscillation. Thus, the absolute ruin probability in the model is decomposed as the sum of two absolute ruin probabilities, where one is the probability that absolute ruin is caused by a claim and the other is the probability that absolute ruin is caused by oscillation.

On absolute ruin minimization under a diffusion approximation model

In this paper, we assume that the surplus process of an insurance entity is represented by a pure diffusion. The company can invest its surplus into a Black–Scholes risky asset and a risk free asset. We impose investment restrictions that only a limited amount is allowed in the risky asset and that no short-selling is allowed. We further assume that when the surplus level becomes negative, the company can borrow to continue financing. The ultimate objective is to seek an optimal investment strategy that minimizes the probability of absolute ruin, i.e. the probability that the liminf of the surplus process is negative infinity. The corresponding Hamilton–Jacobi–Bellman (HJB) equation is analyzed and a verification theorem is proved; applying the HJB method we obtain explicit expressions for the S-shaped minimal absolute ruin function and its associated optimal investment strategy. In the second part of the paper, we study the optimization problem with both investment and proportional reinsurance control. There the minimal absolute ruin function and the feedback optimal investment–reinsurance control are found explicitly as well.

The probabilities of absolute ruin in the renewal risk model with constant force of interest

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.

The probabilities of absolute ruin in the renewal risk model with constant force of interest

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.

Insurance with borrowing: first- and second-order approximations

We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T 1,T 2,…. At the time T η(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(T η(x)).

Insurance with borrowing: first- and second-order approximations

We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T1,T2,…. At the time Tη(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(Tη(x)).