Fractional Distributions and Stochastic Processes with Applications

在运筹学的各个应用领域管理风险的过程中，为随机现象制定灵活而又简约的模型是一个关键因素。在此背景下，Kulkarni类相型分布已被证明具有显著的建模优势和应用。在这个项目中，我们将考虑分数位相型分布的扩展和计数过程的构造，其中到达间隔时间具有上述分布。分数相型分布允许对保险数学中特别关注的尾部相关性进行建模，特别是在重尾现象发生时。此外，还将考虑从分数相位型到分数马尔可夫到达过程的进一步扩展及其应用。

The formulation of flexible and at the same time parsimonious models for stochastic phenomena is a crucial ingredient in the process of managing risks in various application areas of operations research. In this context, Kulkarni class of phase-type distributions has been proven to have significant modelling advantages and applications. In this project we will consider extensions of fractional Phase-type distributions and construction of counting processes where the inter-arrival times have the aforementioned distribution. Fractional phase-type distributions allow to model the tail dependence which is a particular concern in insurance mathematics, especially for when heavy tailed phenomena occur.Applications of this process in the context of insurance will be considered. Additionally, further extensions from fractional phase types to fractional Markov arrival processes and their applications would be considered.