Finite Time Ruin Probability with Variable Interest Rate and Extended Regular Variation

Consider an insurance risk model, in which the surplus process satisfies a recursive equationU n =U n?1(1+r n )?X n forn≥1, whereU 0=x≥0 is the initial surplus, {r n ;n≥1} the interest rate sequence, {X n ;n≥1} the sequence of i. i. d. real-valued random variables with common distribution functionF, which denotes the gross loss during thenth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result asx→∞. Key words variable interest rate - extend regular variation - finite time ruin probability CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)Biography: WEI Xiao (1979-), female, Ph. D candidate, research direction: large deviations and its applications, insurance mathematics.

The Ruin Probability in the Presence of Extended Regular Variation and Optimal Investment

Considering the classical model with risky investment, we are interested in the ruin probability that is minimized by a suitably chosen investment strategy for a capital market index. For claim sizes with common distribution of extended regular variation, starting from an integro-differential equation for the maximal survival probability, we find that the corresponding ruin probability as a function of the initial surplus is also extended regular variation.

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums $S(t) = \sum _{i = t}^{N(t)} X_i ,t \geqslant 0$ , whereX n,n?1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t?0 is a process of non-negative integer-valued random variables, independent ofX n,n?1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t?0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.

Ruin probability of the renewal model with risky investment and large claims

The ruin probability of the renewal risk model with investment strategy for a capital market index is investigated in this paper.For claim sizes with common distribution of extended regular variation,we study the asymptotic behaviour of the ruin probability.As a corollary,we establish a simple asymptotic formula for the ruin probability for the case of Pareto-like claims.

The Asymptotic Behavior of the Ruin Probability within a Random Horizon

Subject to the assumption that the common distribution of claim sizes belongs to the extendedregular variation class,the present work obtains a simple asymptotic formula for the ruin probability within arandom or nonrandom horizon in the renewal model.

Uniform tail asymptotics for the aggregate claims with stochastic discount in the renewal risk models

Considering an insurer who is allowed to make risk-free and risky investments, as in Tang et al.(2010), the price process of the investment portfolio is described as a geometric L′evy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of extended regular variation, we obtain an asymptotically equivalent formula which holds uniformly for all time horizons, and furthermore, the same asymptotic formula holds for the finite-time ruin probabilities. The results extend the works of Tang et al.(2010).

Precise Large Deviations for Sums of Claim-size Vectors in a Two-dimensional Size-dependent Renewal Risk Model

Consider a two-dimensional renewal risk model,in which the claim sizes{Xk;k≥1}form a sequence of i.i.d.copies of a non-negative random vector whose two components are dependent.Suppose that the claim sizes and inter-arrival times form a sequence of i.i.d.random pairs,with each pair obeying a dependence structure via the conditional distribution of the inter-arrival time given the subsequent claim size being large.Then a precise large-deviation formula of the aggregate amount of claims is obtained.

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.

Precise Large Deviations for a Customer-based Individual Risk Model

In this paper, we propose a customer-based individual risk model, in which potential claims by customers are described as i.i.d, heavy-tailed random variables, but different insurance policy holders are allowed to have different probabilities to make actual claims. Some precise large deviation results for the prospectiveoss process are derived under certain mild assumptions, with emphasis on the case of heavy-tailed distribution function class ERV （extended regular variation）. Lundberg type limiting results on the finite time ruin probabilities are also investigated.