  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.
