变利率风险模型有限时间破产概率的渐近

The Asymptotic of Finite Time Ruin Probabilitiesfor Risk Model with Variable Interest Rates

摘要: 本文考虑离散时间风险模型U_n=(U_n-1+Y_n)(1+r_n)-X_n, n=1,2,\cdots, 其中U_0=x>0为保险公司的初始准备金, r_n为在第n个时刻的利率, Y_n为到时刻n为止的总保费收入, X_n为到时刻n为止的所支付的全部索赔, U_n表示保险公司在时刻n的盈余. 当Y_n和r_n满足某些温和条件时, 我们得到了在\, x\to\infty时, 有限时间破产概率\psi(x,N)=\pr\big(\min\limits_0\leq n\leq NU_n<0|U_0=x\big)关于N\geq1的一致渐近的关系式\, \psi(x,N)\sim\tsm_k=1^N\olF_X((1+r_1)\cdots(1+r_n)x), 其中\olF_X(x)是X_1的尾分布.

Abstract: Consider a discrete time risk model \U_n=(U_n-1+Y_n)(1+r_n)-X_n,\qq n=1,2,\cdots, \ where U_0=x>0 is the initial reserve of an insurance company, r_n the interest rates, Y_n the total amount of premiums, X_n the total amount of claims and U_n the reserve at time n. Under some mild conditions on Y_n and r_n, we obtain the uniform asymptotics relation for the finite time ruin probabilities \psi(x,N)\sim\tsm_k=1^N\olF_X((1+r_1)\cdots(1+r_n)x) as x\to\infty, where \psi(x,N)=\pr\big(\min\limits_0\leq n\leq NU_n<0 |U_0=x\big), N\geq1, \olF_X(x) is the tail distribution of X_1, and the uniformity is with respect to N\geq1. \newcommand\fundinfoSupported by the National Natural Science Foundation of China (10671149, 10801139) and Key Project of Philosophy and Social Sciences Research of the Ministry of Education (07JZD0010).