Fractional Distributions and Stochastic Processes with Applications

制定灵活的，同时简约的随机现象模型是一个至关重要的因素，在管理风险的过程中，在各种应用领域的运筹学。在这种情况下，Kulkarni类相型分布已被证明具有显着的建模优势和应用。在这个项目中，我们将考虑分数相位型分布的扩展和计数过程的建设，其中到达时间间隔具有上述分布。分数相位型分布允许模拟尾部依赖性，这是保险数学中特别关注的问题，特别是当重尾现象发生时，将考虑此过程在保险方面的应用。此外，分数相位类型的分数马尔可夫到达过程及其应用的进一步扩展将被考虑。

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.