ABSTRACT

The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite variation stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that for each fixed point in time, $t\ge 0$, a scaling limit of the intensity process converges in distribution, and the limit equals the strong solution of the square-root diffusion process (Cox–Ingersoll–Ross process) at t. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case.

摘要

本文的目的是研究自激跳跃过程的性质，其中强度由 SDE 给出，SDE 由有限变化随机跳跃过程驱动。跳跃前的强度过程值可能会影响跳跃的大小分布。我们关注这个强度函数的性质，并表明对于每个固定时间点，$吨\ge 0$，强度过程的尺度极限在分布中收敛，并且该极限等于平方根扩散过程（Cox-Ingersoll-Ross 过程）在t处的强解。作为一个具体的例子，我们研究了线性强度过程的情况，并在这种情况下推导出期望和方差的显式表达式。