Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order $k$ and beyond

On multivariate modifications of Cramer Lundberg risk model with constant intensities

The paper considers very general multivariate modifications of Cramer-Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes within a type have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer-Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order k, the Poisson negative binomial, the Polya-A…

Fractional discrete processes: compound and mixed Poisson representations

We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.

若干复合分数阶Poisson过程

Some Compound Fractional Poisson Processes

该研究引入并分析三种复合Poisson过程的分数阶版本，证明其为广义计数过程的极限情形，推导了相关统计性质与一维分布的等价形式，丰富了分数阶随机过程的理论研究。

Optimal dividend problem for a generalized compound Poisson risk model

In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Polya-Aeppli risk model as special cases. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that under some conditions the optimal dividend strategy is formed by a barrier strategy.

复合泊松过程的分数阶推广

Fractional Generalizations of the Compound Poisson Process

该研究提出统一的广义分数阶复合泊松过程GFCPP，推导其分布、微分方程及鞅性质，证明多种已有分数阶过程为其特例，还研究了时间变换分类并给出样本路径仿真方法。

马尔可夫链的深度学习：Lyapunov函数、Poisson方程与平稳分布

Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributions

本文提出利用深度学习训练神经网络满足首达分析导出的积分方程，自动构造Lyapunov函数，并扩展用于求解Poisson方程和估计平稳分布，在非紧致状态空间上依然有效。