Ӧ�ø���ͳ�� 2009, 25(5) 499-512 DOI:      ISSN: 1001-4268 CN: 31-1256

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һ��Sparre Andersen����ģ�͵�Gerber-Shiu����
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ժҪ�� ���Ŀ�����һ���������������ʱ��������Erlang($n$)��
Erlang($m$)�Ļ�Ϸֲ���Sparre Andersen����ģ��.
��ҪĿ�����о�Gerber-Shiu����$\phi_\delta(u)$,
����֤����$\phi_\delta(u)$����һ���߽׵Ļ���΢�ַ���,
Ȼ�������˹���Lundberg���̸�������,
�ڴ˻����ϵ�����$\phi_\delta(u)$��������˹�任����֤����$\phi_\delta(u)$����һ�����·���,
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�ؼ����� Sparre   Andersen����ģ��   Erlang($n$)   Gerber-Shiu����   �Ʋ�����.
The Gerber-Shiu Function for a Sparre Andersen\\Risk Model
Fan Qingzhu:Yin Chuancun
Department of Statistics and Finance, School of Business, Shihezi University School of Mathematical Sciences, Qufu Normal University
Abstract: In this paper, we consider a Sparre Andersen risk model in which
the claim inter-arrival distribution is a mixture of an Erlang($n$)
distribution and an Erlang($m$) distribution. Our purpose is to
study the Gerber-Shiu function $\phi_\delta(u)$. First, we show that
$\phi_\delta(u)$ satisfies a higher-order integro-differential
equation, and then we discuss the roots of the generalized
Lundberg's equation. On this basis, we drive the Laplace transform
of $\phi_\delta(u)$ and prove that $\phi_\delta(u)$ satisfies a
renewal equation. Finally, we consider an example.
Keywords: Sparre Andersen risk model   Erlang($n$)   Gerber-Shiu function   ruin probability.
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