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We investigate the problem of minimizing ruin probability by joint decisions of excess-of-loss reinsurance and investment in a financial market. The insurer's reserve is modeled by a diffusion process and may be invested in a financial market consisting of a risky asset with the price process following the geometric Brownian motion and a risk-free asset with a fixed return rate. Borrowing is allowed, but with an interest rate higher than the interest rate of the risk-free investment. Meanwhile, an excess-of-loss reinsurance may be purchased to alleviate the risk of ruin. We apply stochastic control theory and find the optimal strategy of joint reinsurance and investment decisions, and derive the closed form expression of the minimum ruin probability function. Results are illustrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study a Lévy risk process consisting of Brownian component together with premiums and claims that are phase type with many phases. Our aim is to approximate the probability of ruin without touching an upper barrier a. In line with this, the study demonstrates that the described Lévy risk process can essentially be replaced with a simpler risk process in which both premiums and claims are phase-type with just few phases. [ABSTRACT FROM AUTHOR]

Asymptotic Ruin Probabilities in the Cramér-Lundberg model has been widely studied when the claims have light-tailed or heavy-tailed distributions. However, it has not been studied for extreme values over some threshold. In this paper, we investigate the use of the peaks over threshold method, to construct a new estimator of the ruin probability for a risk process with heavy tails claims amounts with infinite variance for the stationary arrival claims in an infinite time. Our approach is based on approximating the sample over some threshold by the generalized Pareto distribution. We prove that the proposed estimator is consistent and asymptotically normal. The performance of our new estimator is illustrated by some results of simulations for some loss models and provides an extensive example application to Danish data on large fire insurance. [ABSTRACT FROM AUTHOR]

Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations. [ABSTRACT FROM AUTHOR]

Parisian ruin occurs once the surplus stays continuously below zero for a given period. We consider the spectrally negative Lévy risk process where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially or mixed Erlang distributed random variable, or once it reaches a given negative threshold. We consider the Laplace transform of the time to ruin and the Laplace transform of the time that the process is negative. [ABSTRACT FROM AUTHOR]

The paper overviews stochastic optimization models of actuarial mathematics and methods for their solution from the point of view of the methodology of multicriteria stochastic programming and optimal control. The evolution of the capital of an insurance company is considered in discrete time. The main random parameters of the models are insurance payouts, i.e., the ratios of paid insurance claims to the corresponding premiums per unit time. Optimization variables are the structure of the insurance portfolio (gross premium structure) and amount of dividends. As efficiency criteria, indicators of the profitability of the insurance business are used, and, as risk indicators the ruin probability and the recourse capital necessary to prevent the ruin are taken. The goal of the optimization is to find Pareto-optimal solutions. Methods for finding these solutions are proposed. [ABSTRACT FROM AUTHOR]

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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.

We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (Scand Actuar J 2000(1):46–62, 2000; J Appl Probab 41(2):570–578, 2004; J Appl Probab 43:535–551, 2006) and Lefèvre and Loisel (Methodol Comput Appl Probab 11(3):425–441, 2009) for the case of stationary Poisson claim arrivals and by Lefèvre and Picard (Insurance Math Econom 49:512–519, 2011; Methodol Comput Appl Probab 16:885–905, 2014), for OS claim arrivals. [ABSTRACT FROM AUTHOR]

The Cramer-Lundberg model is considered as a model of insurance company. Since it is impossible to obtain an explicit solution for the non-ruin probability function of insurance company for an arbitrary distribution of the values of insurance claims, the authors consider the problem of estimating the convergence of the original non-ruin probability to one that would be obtained by approximating the values of claim distribution function. [ABSTRACT FROM AUTHOR]

We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields new expressions for the ruin probability in the insurance and the dual risk models with dependence between the claim severities or capital gains respectively. [ABSTRACT FROM AUTHOR]

The approximations of risk processes with mixed exponentially distributed inter-arrival times are investigated. The number of claims in a fixed time interval is Mixed Poisson distributed. The approximating process is always overdispersed. This allows a better fit to more realistic situations in finances, than e.g. classical Cramér-Lundberg model. The claim sizes are divided in three different groups, dependently on finiteness of their first two moments. We illustrate all the cases by numerical examples. The case of diffusion approximation is investigated. Both American and European Pareto claims sizes are also studied. [ABSTRACT FROM AUTHOR]

We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell–Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell–Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g. those obtained for the case of stationary Poisson claim arrivals. [ABSTRACT FROM PUBLISHER]

In a virtual organization directed on the insurance business, the estimations of the risk process and of the ruin probability are important concerns: for researchers, at the theoretical level, and for the management of the company, as these influence the insurer strategy. We consider the evolution over an extended period of time of the insurer surplus process. In this paper, we present some methods for the estimation of the ruin probability and for the evaluation of a reserve fund. We discuss the ruin probability with respect to: the parameters of the individual claim distribution, the load factor of premiums and the intensity parameter of the number of claims process. We analyze the model in which the premiums are computed according to the mean value principle. Also, we attempt the case when the initial capital is proportional to the expected value of the individual claim. We give numerical illustration.

The paper deals with a ruin problem, where there is a Parisian delay and a lower ultimate bankrupt barrier. In this problem, we will say that a risk process get ruined when it stays below zero longer than a fixed amount of timeζ> 0 or goes below a fixed level −a. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we identify the Laplace transform of the ruin probability in terms of so-called q-scale functions. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples. [ABSTRACT FROM AUTHOR]

We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields new expressions for the ruin probability in the insurance and the dual risk models with dependence between the claim severities or capital gains respectively.

In this paper, we consider dividend problem for an insurance company whose risk evolves as a spectrally negative Lévy process (in the absence of dividend payments) when a Parisian delay is applied. An objective function is given by the cumulative discounted dividends received until the moment of ruin, when a so-called barrier strategy is applied. Additionally, we consider two possibilities of a delay. In the first scenario, ruin happens when the surplus process stays below zero longer than a fixed amount of time. In the second case, there is a time lag between the decision of paying dividends and its implementation. [ABSTRACT FROM AUTHOR]

Problems of computational actuarial mathematics, dynamic financial analysis, and optimization of insurance business and the possibility of their solution by means of parallel computing on graphics accelerators are discussed. The ruin probability and other performance criteria of an insurance company are estimated by the Monte Carlo method. In many cases, it is the only applicable method. Since the ruin probability is small enough, to achieve an acceptable estimate accuracy, an astronomical number of simulations may be required. Parallelization of the Monte Carlo method and the use of graphical accelerators allow us getting the desired result in a reasonable time. The results of numerical experiments on the developed system of actuarial modeling are presented, allowing the use of graphical accelerator that supports Nvidia CUDA 1.3 and higher. [ABSTRACT FROM AUTHOR]

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples. [ABSTRACT FROM AUTHOR]

We extend the classical balloting problem to the case of two random streams. The procedure has applications in risk theory. [ABSTRACT FROM AUTHOR]

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included. [ABSTRACT FROM AUTHOR]

A stochastic successive approximation method is analyzed with a view to solving risk assessment problems that are reduced to a renewal integral equation and, in particular, to assessing the insolvency risk of an insurance company. Integrals in the equation are evaluated approximately, for example, by the Monte Carlo method. Iterations of the method are proved to converge uniformly with probability one. Theoretical results are illustrated by numeral computations. [ABSTRACT FROM AUTHOR]

In this paper we study an optimal investment problem of an insurer when the company has the opportunity to invest in a risky asset using stochastic control techniques. A closed form solution is given when the risk preferences are exponential as well as an estimate of the ruin probability when the optimal strategy is used. [ABSTRACT FROM AUTHOR]

A Markovian risk process is considered in this paper, which is generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process { N(t)} t≥0 with N(t) being the number of jumps during the interval (0, t] for a Markov jump process. The ruin probability ψ( u) of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ( u) is obtained and the bounds for the convergence rate of the ruin probability ψ( u) are given by using a generalized renewal technique developed in the paper. [ABSTRACT FROM AUTHOR]

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by a Markov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so-called Lundberg estimate for the ruin probabilities and present a numerical example. [ABSTRACT FROM AUTHOR]

Abstract: We consider an insurance company in the case when the premium rate is a bounded non-negative random function and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a and volatility . If we find exact the asymptotic upper and lower bounds for the ruin probability as the initial endowment u tends to infinity, i.e. we show that for sufficiently large u. Moreover if with we find the exact asymptotics of the ruin probability, namely . If , we show that for any . [Copyright &y& Elsevier]

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a risk process with stochastic return on investments. The basic risk process is the classical risk process while the return on the investment generating process is a compound Poisson process plus a Brownian motion with positive drift. We obtain an integral equation for the ultimate ruin probability which is twice continuously differentiable under certain conditions. We then derive explicit expressions for the lower bound for the ruin probability. We also study a joint distribution related to exponential functionals of Brownian motion which is required in the derivations of the explicit expressions for the lower bound. [Copyright &y& Elsevier]

A general risk model that allows for stochastic return on investments as well as perturbation by diffusion is studied. Integro-differential equations for the distributions of the time of ruin, the surplus prior to ruin and the deficit at ruin of this model are established. In particular, we consider a diffusion perturbed risk model with interest force in details. [ABSTRACT FROM AUTHOR]

In this paper, a class of risk processes perturbed by diffusion are considered. The Lundberg inequalities for the ruin probability are obtained. The size of the Lundberg exponents for different kinds of risk model is compared. The numerical illustration for the impact of the parameters on the ruin probability is given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]