Yu Jinyou,Hu Yijun,Wei Xiao
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2010, 26(1): 57-65.
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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
$\psi(x,N)\sim\tsm_{k=1}^{N}\ol{F}_X((1+r_1)\cdots(1+r_n)x)$ as
$x\to\infty$, where $\psi(x,N)=\pr\big(\min\limits_{0\leq n\leq
N}U_n<0$ $|U_0=x\big)$, $N\geq1$, $\ol{F}_X(x)$ is the tail
distribution of $X_1$, and the uniformity is with respect to
$N\geq1$.}
\newcommand{\fundinfo}{Supported 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).