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On a class of Erlang(2) risk processes. (Chinese. English summary) Zbl 1098.91077
Summary: The authors consider a class of actuarial risk models when the waiting times have an Erlang(2) distribution, the same distribution as Gamma(2). This generalizes the classical model, where the waiting times are exponential, and gives more flexibility in the modelling of a risk business. Firstly, the authors prove that the survival probability $$R(u)$$ fulfills an integro-differential equation. Then an exponential-type integral equation satisfied by the survival probability $$R(u)$$ is obtained. Finally, an explicit solution for the survival probability $$R(u)$$ is obtained in a closed form.
This work is a continuation and supplement of the important corresponding work of D. C. M. Dickson N. Am. Actuar. J. 2, No. 3, 60-73 (1998; Zbl 1081.60549)] and D. C. M. Dickson and C. Hipp [Insur. Math. Econ. 22, No. 3, 251–262 (1998; Zbl 0907.90097); ibid. 29, No. 3, 333–344 (2001; Zbl 1074.91549)].
##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60E05 Probability distributions: general theory 62E17 Approximations to statistical distributions (nonasymptotic)