论文标题
自我激发的跳跃过程及其渐近行为
Self-exciting jump processes and their asymptotic behaviour
论文作者
论文摘要
本文的目的是研究自我激发跳跃过程的属性。我们得出了具有独立的，相同分布的跳跃大小的SDE驱动的自兴趣过程的拉普拉斯变换。通过使用这种拉普拉斯变换，我们找到了一个自我激发过程时刻的递归公式。瞬间的公式使我们能够为自我激发过程的期望和差异得出表达。我们表明，自我兴奋的过程可以表现出有限和无限的活动行为。此外，我们表明，强度过程的缩放限量等于分布中平方根扩散过程（Cox-Ingersoll-ross过程）的强溶液。作为一个特殊的例子，我们研究了线性强度过程的情况，并在这种情况下为期望和方差得出明确的表达式。
The purpose of this paper is to investigate properties of self-exciting jump processes. We derive the Laplace transform of SDE driven self-exciting processes with independent, identically distributed jump sizes. By using this Laplace transform, we find a recursive formula for the moments of the self-exciting process. The formula for the moments allow us to derive expressions for the expectation and variance of the self-exciting process. We show that self-exciting processes can exhibit both finite and infinite activity behaviour. Furthermore, we show that the scaling limit of the intensity process equals the strong solution of the square-root diffusion process(Cox-Ingersoll-Ross process) in distribution. 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.
