若干复合过程的积分
Integrals of some compound processes
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中文总结 AI 辅助
该研究基于复合泊松过程理论构建通用框架，证明伯恩斯坦从属的时间变换性质，推导复合过程积分的显式表示等结果，为相关随机过程研究提供理论支撑。
中文摘要 AI 辅助
利用复合（非齐次）泊松过程理论，我们定义了一个通用框架，涵盖泊松过程的多数推广形式，包括Skellam意义下和空间分数阶意义下的推广。我们证明，伯恩斯坦从属是唯一能导出复合泊松过程的时间变换方式。还考虑了带有逆伯恩斯坦从属子的从属情形。接着聚焦复合过程的积分，给出复合泊松过程分数阶积分的显式表示，证明固定t时，该积分服从复合泊松随机变量分布，并将此结果推广到更一般的积分形式。此外，得到一些极限结果、迭代积分的显式形式及其控制方程，最后在傅里叶-拉普拉斯域研究复合更新过程的积分。
英文摘要
By means of the theory of compound (non-homogenoeus) Poisson processes we define a general framework which includes most of the generalizations of the Poisson processes, both in the Skellam sense and in the space-fractional sense. We prove that Bernstein subordination is the only time-changing leading to a compound Poisson process. We also consider the case of subordination with inverse Bernstein subordinators. Then, we focus on the integrals of compound processes. We give an explicit representation of the fractional integral of compound Poisson process, showing that for fixed t it is distributed as a compound Poisson random variable. We then extend this result to more general integral forms. Furthermore, we obtain some limit results, explicit forms of the iterated integrals and their governing equation. Finally, we study the integral of compound renewal processes in the Fourier-Laplace domain.
