Local Theorem For Ruin Probability In Risk Model

The asymptotic estimate for ruin probability, which have the important guideeffect in practice, is one of the most important topics in risk theory. People havehad perfect investigation for the classical Cramér-Lundberg risk model already.Embrechts and Varaverbeke investigated the renewal risk model and gave a tailequivalence relationship of the ruin probabilityψ(x), i.e.ψ(x)～1/ρ(?)_e(x), under theassumption that the claim size is heavy-tailed, which is regard as a classical result inthe context of extremal value theory. We define R(x, x+z]=ψ(x)-ψ(x+z), whichis the local result for ruin probability. Nowadays, people are very interested in theasymptotical property for ruin probability or what is asymptotically proportional towhen x trend to infinite.Moreover, the risk model in reality is not ideal as the classical risk model. Per-turbed risk models were introduced by Gerber (1970) who extended the classicalCramér-Lundberg risk model by adding a Brownial motion. The perturbed riskmodels described the reality very much to the point.In this paper, we consider the local theorem for ruin probability in risk theory. Weassume that the relative safety loading condition i.e.ρ＞0 is hold all through the paper.First, we investigate the local theorem for ruin probability with the distribution funtionof claim size F is belong to S^* in the perturbed renewal risk model and perturbedstationary renewal risk model. Then, we have a investigation into the local theoremfor ruin probability in Cramér-Lundberg risk model, renewal risk model, stationaryrenewal risk model and delayed renewal risk model under the assumption that the claimsize F belong to S^*(ν). Finally, we investigate the Cramér-Lundberg risk modelpertured by diffusion and obtain a local asymptotic expression for ruin probabilitywhen the claim size F belong to S^*(ν). From that result we can see the influence ofdiffusion can not be neglected under the situation.The following are main results of this thesis:1.Local theorem based on S^* for ruin probability in perturbed renewalrisk model:In the perturbed renewal risk model with relative safety loading conditonρ＞0,if the distribution function of claim size F belong to S^*, then for all z＞0, R(x,x+z]～z/(ρμ)(?)(x)2.Local theorem based on S^* for ruin probability in perturbed station-ary renewal risk model: In the perturbed stationary renewal risk model with relative safety loading con-ditonρ＞0, if the distribution function of claim size F belong to S^*, then for allz＞0, R(x,x+z]～z/(ρμ)(?)(x)3.Local theorem based on S^* (ν) for ruin probability in Cramér-Lundbergrisk model,In the perturbed stationary renewal risk model with relative safety loading con-ditonρ＞0, if the distribution function of claim size F belong to S^*, then for all z＞0,and integral from n=0 to∞e^νt(?)(t)＜c/λ, i.e. the Lundberg exponent is not exist, then for all z＞0, lim_{x→∞} (R(x,x+z])/((?)(x))=(q(1-q)(1-e^-νz))/(μν(1-((1-q)/μ) integral from n=0 to∞e^νt(?)(t)dt)²)4. Local theorem based on S^*(ν) for ruin probability in renewal riskmodel:In the renewal risk model with relative safety loading conditonρ＞0, if thedistribution function of claim sizeF∈S^*(ν),ν＞0, and integral from n=0 to∞e^-νctdG(t)＜∞, the forall z＞0, lim_{x→∞} (R(x,x+z])/((?)(x))=z/(ρμ) integral from n=0 to∞e^-νctdG(t)5.Local theorem based on S^*(ν) for ruin probability in stationary re-newal risk model:In the stationary renewal risk model with relative safety loading conditonρ＞0,if the distribution function of claim size F∈S^*(ν),ν＞0, the for all z＞0, lim_{x→∞}(R(x,x+z])/((?)(x))=(1-e^-νz)/((1+ρ)+μν)+z/(ρ(1+ρ)μ) integral from n=0 to∞e^-νctdG(t)6.Local theorem based on S^*(ν) for ruin probability in delayed renewalrisk model:In the delayed renewal risk model with relative safety loading conditonρ＞0,and integral from n=0 to∞e^-νctdG(t)＜∞, if the distribution function of claim size F∈S^*(ν), (ν＞0)then for all z＞0, lim_{x→∞} (R(x,x+z])/((?)(x))= z/(ρμ) integral from n=0 to∞e^-νctdG(t)7. Local theorem based on S^* (ν) for ruin probability in perturbed Cramér-Lundberg risk model: In the Cramér-Lundberg risk model with relative safety loading conditonρ＞0,we assume that c/D＞ν, if the distribution function of claim size F∈S^*(ν), andintegral form n=0 to∞e^νy(?)(y)dy＜(c-Dν)/λ, then for all z＞0 lim_{x→∞} (R(x,x+z])/((?)(x))=(q(1-q))/(μν)(1-e^-νz)((c/D)/(c/D-ν))² 1/((1-(λ/(c-Dν)) integral from n=0 to∞e^νy(?)(y)dy)²)...