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

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).