Fractional Distributions and Stochastic Processes with Applications

在运筹学的各个应用领域中，建立灵活而又简洁的随机现象模型是风险管理过程中的一个关键因素。在这种情况下，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尾依赖性进行建模，这在保险数学中是一个特别关注的问题，特别是当出现重尾现象时。将考虑这一过程在保险方面的应用。此外，从分数相类型进一步扩展到分数马尔可夫到达过程及其应用将被考虑。

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.