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Joint distributions of some actuarial random vectors in the continuous-time compound binomial model. (Chinese. English summary) Zbl 1240.91090

Summary: The continuous-time compound binomial model, firstly proposed by G. Liu, Y. Wang and B. Zhang [Insur. Math. Econ. 36, No. 3, 303–316 (2005; Zbl 1110.62146)], is the continuous-time version of the compound binomial model. In this paper, a renewal mass function of a defective renewal sequence constituted by the up-crossing zero points is introduced in the continuous-time compound binomial model. By the mass function together with the strong Markov property of the surplus process \(X(t)\), explicit expressions of the ruin probability and the joint distributions of some actuarial random vectors such as \((T,X(T-),|X(T)|)\), \((T,X(T-),|X(T)|,\inf\limits_{0\leq t <L}X(t))\) and \((T,X(T-),|X(T)|,\sup\limits_{0\leq t<T}X(t))\) are obtained, where \(T\) represents the time of ruin and \(L\) the time of the surplus process leaving deficit ultimately. The corresponding joint distributions are directly obtained for the compound binomial model, \(\{X(n)\}\), as the 1-skeleton chain of the continuous-time compound binomial model. Finally, a special case with the claim amount being geometrically distributed in the compound binomial model is considered.

MSC:

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 1110.62146
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