Bankruptcy, A Few Kind Of Extended Risk Model

This dissertation is devoted to dealing with ruin theory for some kinds of risk models which include the Sparre Andersen risk model, Cox risk model and the generalized Cox risk model, and we discuss the Cox risk model with stochastic rates of interest.Sparre Andersen considered the situation in which claims occur as a general renewal process in 1957, then he constructed the renewal risk model and began to study ruin probability. Since then, the calculation of ruin probability became increasingly important. See, [2] [3] [4] [15] for details. In [5] a clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount is exponential distribution. Prom [1], we know the result of the case when the claim amount is the mixture of two exponentials. In Chapter 1, a clear expression for Laplace transform of the finite time ruin probability is given when the claim amount distribution is a mixture of finite exponential distributions, Theorem 1.2.1 Let the claim sizes {Y_i,i ≥ 1} and interoccurrence times {T_i,i ≥ 1} be mutually independent and i.i.d. Let Y_i be a finite mixed exponential distribution and its p.d.f be ∑_i=1ⁿ A_iλ_ie^-λ_iy, y > 0, 0 < λ₁ < λ2 < … <λ_n, ∑_i=1ⁿ A_i=1 , and A_i ≥ 0, thenwhere J_i is given by (1.2.10).Considering the limitations of classical model with single-type-insurance, Zhim-ing Jiang et al.[40] construct multitype-insurance model and study the ruin probability of a class of twotype-insurance model. Considering the varying of claims intensity, in Chapter 2, firstly we study a multitype-insurance model in which claim processes containing compound Poisson process and Cox process. We obtain the expression of ruin probability under influence of variable premium. Theorem 2.2.1 Under the above assumptions and if c(r) >αμ₁ + βμ₂ for allr≥ 0, let F = 1 - F be the tail distribution of F, then andTheorem 2.2.2 Under the above assumptionsandIn Section 2.4 we extend the risk model in Section 2.2 to a generalized model, in which two independent Cox risk process construct the claim process. We get Theorem 2.4.1 Let F = 1 — F. Under the above assumptions and if c(r) > αμ₁ + βμ₂ for all r ≥ 0, thenand Theorem 2.4.2 Under the above assumptionsand Recently there are a lot of papers considering a risk process with stochastic interest rate, see, [10] [12] [14] [15] [24] [25] for details. Jun Cai [12] deals with a compound Poisson process invested in a stochastic interest process. Using a differential argument, an integro-differential equation satisfied by the penalty function is given. Chapter 3 is a investigation into the ruin problem in Cox risk model with stochastic rates of interest. Using a differential argument, we derive an integro-differential equation satisfied by the penalty function. Theorem 3.2.1 Let stochastic interest process R_t = δt + σB_t, where B_t is a standard Brownian motion. The distribution of claims amount is F(y), which have a density function f(y). Suppose φ_ir^'''(u) is continuous in u ≥ 0 and both φ_ir^'(u), φ_ir^''(u), and φ_ir^''(u) are bounded in u ≥ 0. Then φ_ir(u) satisfies the following integro-differential equation:...