Types Of Bankruptcy Risk Model Theory,

This dissertation is devoted to the development of ruin theory in risk models. To begin with, under the assumption that the claim size is heavy-tailed, we discuss some tail equivalence relationships of (x) in several risk models. We also consider local limit theorems for ruin probability in the classical risk model perturbed by diffusion and the Erlang(n, β) risk model.In Chapter 1, in order to make the risk model more realistic, we extend the Sparre Andersen risk model and introduce the following risk models.Model 1 The claim sizes {Zi,i ≥ 1} form a sequence of i.i.d. and non-negative r.v.s. with common distribution F and finite mean EZ. The inter-occurrence times {θi,i≥1} are independent and non-negative r.v.s., where 1,2, ...θm (1 ≤m <θ) have distributions G1, G2, . . . , Gm, respectively and {θi,i > m} have common distribution G (G is different from G1, G2, ..., Gm) with finite mean Ed. Define a counting process N(t) = sup {n≥1|Tn (0, t]} ,where Tn = 1 and N(t) is independent of {Zi,i≥1}. Therefore thecorresponding risk process S(t) is defined by S(t) .Assume that therelative safety loading condition holds.Model 2 The claim sizes {Zi,i ≥1} form a sequence of independent and non-negative r.v.s., where Z1,Z2,...,Zn (1 ≤n <θ) have distributions F1,F2,... ,Fn, respectively and {Zi,i >n} have common distribution F (F is different from F1,F2, . . . , Fn) with finite mean EZ. Assume that the inter-occurrence times {θi,i ≥1} and N(t) are the same as defined in Model 1. Thusthe corresponding risk process S(t) is defined by . Further, weassume that relative safety loading condition p = 0 holds.Model 3 Define the risk process + X(t), where ct is the same as defined in Model 2 and the perturbation process{X(t)} is astochastic process with locally bounded trajectories and X(0)=0, independentN(t)of 2 Zi. Further, we assume that the relative safety loading condition p = holds.Model 4 On the basis of Model 3, define the risk process S(t)where M(t) is a renewal process , i.e. the inter-occurrence times {Yi,i > 1} are i.i.d. non-negative r.v.s. with common distribution Q and finite mean EY. The claim sizes {Xi,i > 1} are different from {Zi,i > 1}, having common distribution P and finite mean EX. Further, assume thatare independent mutually and the relative safetyloading conditionholds.Further, we investigate the tail equivalence relationships for the ruin probabilities in the four models that are introduced above.Theorem 1.3.1. In Model 1 with the relative safety loading condition , thenTheorem 1.3.2. In Model 2 with the relative safety loading condition holds, where Ci are non-negativeconstants, i = 1, 2, . . . , n, then we haveTheorem 1.3.3. In Model 2 with the relative safety loading condition p > 0, if F and holds, where C, are non-negative constants, i = 1, 2, . . . , n, then we haveTheorem 1.3.4. In Model 3 with the relative safety loading condition p > 0, assume that F P or Fe S and Fi(x) ~ CiF(x) holds, where Ci are non-negative constants, i = 1, 2, . . . , n. Further assume that (1) If X(t) =B(t), where is a constant and {B(t),t > 0} is a standard Brownian motion with B(0) = 0;(2) If dX(t) = -aX(t)dt + dB(t),X(0) = 0 with a > 0 and B(t) a standard Brownian motion;then the tail equivalence relationshipholds in the both cases.Theorem 1.3.5. In Model 3 with the relative safety loading condition p > 0, assume that holds, where Ci are non-negative constants, i = 1, 2,..., n. Further assume that and Fe with finite mean mi.(1) If X(t) = aB(t), where a 0 is a constant and {B(t),t > 0} is a standard Brownian motion with .0(0) = 0;(2) If dX(t) = -aX(t)dt + dB(t),X(0) = 0 with a > 0 and B(t) a standard Brownian motion; then the tail equivalence relationshipholds in the both cases.Theorem 1.3.6. In Model 4 with the relative safety loading condition p > 0, assume that holds, where Ci are non-negative constants, z = 1, 2, . . . , n. Further assume that E j < , j = 1, 2, . . . , m and there exists such that the equa...