宇世航,王德輝,魏蘊波
(1.齊齊哈爾大學 理學院,黑龍江 齊齊哈爾 161006;2.吉林大學 數學學院,長春 130012)
考慮基于下述條件的一類風險模型:
Nk=α°Nk-1+εk,k=1,2,…,
(1)
盈余動態過程模型為
(2)
其中:u表示保險公司初始資金;c表示單位收益率;Sk表示從第0到第k期累積索賠總額,即
Sk=W1+W2+…+Wk,
(3)
規定S0=0.
證明: 注意FY∈C,對任意的θ>0和充分大的y,
因此,有
(4)
由引理1和式(4),存在某個δ>0,有
證明:對任意的ε>0,當n→∞時,有
證明: 對任意實值r,Sk的特征函數為
另一方面,有
證明: 令{Y,Yj,j≥1}是與X同分布的i.i.d.序列,且與式(1)中定義的{Nk}獨立.對任意的η>0,考慮
由引理1,對充分大的x,有
(6)
對x>γk一致成立.
(7)
另一方面,
(9)
令η↓0,由式(7),(9)和定理1可證結論.
證明: 事實上,由定理1,只需證存在某個δ>0,使得
(10)
在x>γk1+δ上一致成立即可.類似定理2的證明,只需將引理2應用到式(5),(8)中即可證得式(10).
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