Locally and globally uniform approximations for ruin probabilities of a nonstandard bidimensional risk model with subexponential claims

Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.

UNIFORM ESTIMATE ON FINITE TIME RUIN PROBABILITIES WITH RANDOM INTEREST RATE

We consider a discrete time risk model in which the net payout （insurance risk） {Xk, k = 1, 2,...} are assumed to take real values and belong to the heavy-tailed class L∩ D and the discount factors （financial risk） {Yk, k = 1,2,...} concentrate on [θ, L], where 0 〈 0 〈 1, L 〈 ∞, {Xk, k = 1,2,...}, and {Yk, k=1,2,...} are assumed to be mutually independent. We investigate the asymptotic behavior of the ruin probability within a finite time horizon as the initial capital tends to infinity, and figure out that the convergence holds uniformly for all n ≥ 1, which is different from Tang Q H and Tsitsiashvili G （Adv Appl Prob, 2004, 36： 1278-1299）.

CALCULATIONS OF RUIN PROBABILITIES CONCERNING WITH CLAIM OCCURRENCES

In this article, we consider the perturbed classical surplus model. We study the probability that ruin occurs at each instant of claims, the probability that ruin occurs between two consecutive claims occurrences, as well as the distribution of the ruin time that lies in between two consecutive claims. We give some finite expressions depending on derivatives for Laplace transforms, which can allow computation of the probabilities concerning with claim occurrences. Further, we present some insight on the shapes of probability functions involved.

Uniform Asymptotics for Finite-Time Ruin Probabilities of Risk Models with Non-Stationary Arrivals and Strongly Subexponential Claim Sizes

This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process.Under the assumption that the claim-number process satisfies the large deviations principle,the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes.Further,as an application of the result of onedimensional risk model,we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.

Ruin Probabilities under a Markovian Risk Model

In this paper, a Markovian risk model is developed, in which the occurrence of the claims is described by a point process {N(t)} t≥0 with N(t) being the number of jumps of a Markov chain during the interval [0, t]. For the model, the explicit form of the ruin probability Ψ(0) and the bound for the convergence rate of the ruin probability Ψ(u) are given by using the generalized renewal technique developed in this paper. Finally, we prove that the ruin probability Ψ(u) is a linear combination of some negative exponential functions in a special case when the claims are exponentially distributed and the Markov chain has an intensity matrix (q ij ) i,j∈E such that q m = q m1 and q i = q i(i+1), 1 ≤ i ≤ m−1.

Ruin probabilities with insurance and financial risks having an FGM dependence structure

We consider a discrete-time risk model,in which insurance risks and financial risks jointly follow a multivariate Farlie-Gumbel-Morgenstern distribution,and the insurance risks are regularly varying tailed.Explicit asymptotic formulae are obtained for finite-time and infinite-time ruin probabilities.Some numerical results are also presented to illustrate the accuracy of our asymptotic formulae.

Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.

Ruin Probabilities for a Two-Dimensional Perturbed Risk Model with Stochastic Premiums

In this paper, we consider a two-dimensional perturbed risk model with stochastic premiums and certain dependence between the two marginal surplus processes. We obtain the Lundberg-type upper bound for the infinite-time ruin probability by martingale approach, discuss how the dependence affects the obtained upper bound and give some numerical examples to illustrate our results. For the heavy-tailed claims case, we derive an explicit asymptotic estimation for the finite-time ruin probability.

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d.standard uniform random variables.This extends a corresponding result of Korshunov.As an application,we generalize a result of Tang,the uniform asymptotic estimate for the finite-time ruin probability,to the whole strongly subexponential class.

Ruin Probabilities in the Risk Process with Random Income

We extend the classical risk model to the case in which the premium income process, modelled as a Poisson process, is no longer a linear function. We derive an analog of the Beekman convolution formula for the ultimate ruin probability when the inter-claim times are exponentially distributed. A defective renewal equation satisfied by the ultimate ruin probability is then given. For the general inter-claim times with zero-truncated geometrically distributed claim sizes, the explicit expression for the ultimate ruin probability is derived.

Upper Bounds for Ruin Probabilities under Stochastic Interest Rate and Optimal Investment Strategies

In this paper, we study the upper bounds for ruin probabilities of an insurance company which invests its wealth in a stock and a bond. We assume that the interest rate of the bond is stochastic and it is described by a Cox-Ingersoll-Ross （CIR） model. For the stock price process, we consider both the case of constant volatility （driven by an O U process） and the case of stochastic volatility （driven by a CIR model）. In each case, under certain conditions, we obtain the minimal upper bound for ruin probability as well as the corresponding optimal investment strategy by a pure probabilistic method.

Ruin Probabilities for a Risk Model with Two Classes of Claims

In this paper we consider a risk model with two kinds of claims, whose claims number processes are Poisson process and ordinary renewal process respectively. For this model, the surplus process is not Markovian, however, it can be Markovianized by introducing a supplementary process, We prove the Markov property of the related vector processes. Because such obtained processes belong to the class of the so-called piecewise-deterministic Markov process, the extended infinitesimal generator is derived, exponential martingale for the risk process is studied. The exponential bound of ruin probability in iafinite time horizon is obtained.

Ruin Probabilities in Cox Risk Models with Two Dependent Classes of Business

In this paper we consider risk processes with two classes of business in which the two claim-number processes are dependent Cox processes. We first assume that the two claim-number processes have a two-dimensional Markovian intensity. Under this assumption, we not only study the sum of the two individual risk processes but also investigate the two-dimensional risk process formed by considering the two individual processes separately. For each of the two risk processes we derive an expression for the ruin probability, and then construct an upper bound for the ruin probability. We next assume that the intensity of the two claim-number processes follows a Markov chain. In this case, we examine the ruin probability of the sum of the two individual risk processes. Specifically, a differential system for the ruin probability is derived and numerical results are obtained for exponential claim sizes.

ASYMPTOTICS FOR RUIN PROBABILITIES OF TWO KINDS OF DEPENDENT RISK MODELS WITH NLOD INTER-ARRIVAL TIMES

This paper establishes some asymptotic formulas for the infinite-time ruin probabilities of two kinds of dependent risk models. One risk model considers the claim sizes as a modulated process, and the other deals with negatively upper orthant dependent claim sizes. In the two models, the inter-arrival times are both assumed to be negatively lower orthant dependent.

Explicit Expressions for the Ruin Probabilities of Erlang Risk Processes with Pareto Individual Claim Distributions

In this paper we first consider a risk process in which claim inter-arrival times and the time until the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate ruin, given an initial reserve of u when the claim size follows a Pareto distribution. Follow Ramsay[8], Laplace transforms and exponential integrals are used to derive the solution, which involves a single integral of real valued functions along the positive real line, and the integrand is not of an oscillating kind. Then we show that the ultimate ruin probability can be expressed as the sum of expected values of functions of two different Gamma random variables. Finally, the results are extended to the Erlang(n) case. Numerical examples are given to illustrate the main results.

Ruin Probabilities of a Surplus Process Described by PDMPs

In this paper we mainly study the ruin probability of a surplus process described by a piecewise deterministic Markov process （PDMP）. An integro-differential equation for the ruin probability is derived. Under a certain assumption, it can be transformed into the ruin probability of a risk process whose premiums depend on the current reserves. Using the same argument as that in Asmussen and Nielsen, the ruin probability and its upper bounds are obtained. Finally, we give an analytic expression for ruin probability and its upper bounds when the claim-size is exponentially distributed.

Asymptotic Ruin Probabilities of the Renewal Model with Constant Interest Force and Dependent Heavy-tailed Claims

In this paper, we investigate the asymptotic behavior for the finite- and infinite-time ruin probabilities of a nonstandard renewal model in which the claims are identically distributed but not necessarily inde- pendent. Under the assumptions that the identical distribution of the claims belongs to the class of extended regular variation （ERV） and that the tails of joint distributions of every two claims are negligible compared to the tails of their margins, we obtain the precise approximations for the finite- and infinite-time ruin probabilities.

Uniform asymptotics for finite-time ruin probability in some dependent compound risk models with constant interest rate

Consider two dependent renewal risk models with constant interest rate. By using some methods in the risk theory, uniform asymptotics for finite-time ruin probability is derived in a non-compound risk model, where claim sizes are upper tail asymptotically independent random variables with dominatedly varying tails, claim inter-arrival times follow the widely lower orthant dependent structure, and the total amount of premiums is a nonnegative stochastic process. Based on the obtained result, using the method of analysis for the tail probability of random sums, a similar result in a more complex and reasonable compound risk model is also obtained, where individual claim sizes are specialized to be extended negatively dependent and accident inter-arrival times are still widely lower orthant dependent, and both the claim sizes and the claim number have dominatedly varying tails.

RUIN PROBLEM FOR A CLASS OF RISK PROCESSES PERTURBED BY DIFFUSION

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.

RUIN PROBABILITY IN THE CONTINUOUS-TIME COMPOUND BINOMIAL MODEL WITH INVESTMENT

This article deals with the problem of minimizing ruin probability under optimal control for the continuous-time compound binomial model with investment. The jump mechanism in our article is different from that of Liu et al [4]. Comparing with [4], the introduction of the investment, and hence, the additional Brownian motion term, makes the problem technically challenging. To overcome this technical difficulty, the theory of change of measure is used and an exponential martingale is obtained by virtue of the extended generator. The ruin probability is minimized through maximizing adjustment coefficient in the sense of Lundberg bounds. At the same time, the optimal investment strategy is obtained.