Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

In this paper, we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations.Comment: 14 page

A State-Dependent Dual Risk Model

In a dual risk model, the premiums are considered as the costs and the claims are regarded as the profits. The surplus can be interpreted as the wealth of a venture capital, whose profits depend on research and development. In most of the existing literature of dual risk models, the profits follow the compound Poisson model and the cost is constant. In this paper, we develop a state-dependent dual risk model, in which the arrival rate of the profits and the costs depend on the current state of the wealth process. Ruin probabilities are obtained in closed-forms. Further properties and results will also be discussed.Comment: 20 pages, 4 figure

Asymptotic Analysis for Optimal Dividends in a Dual Risk Model

The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for the dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividends problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.Comment: 23 page