In this paper we introduce a new model named CARMA(p,q)-Hawkes process as the Hawkes model with exponential kernel implies a strictly decreasing behaviour of the autocorrelation function and empirically evidences reject the monotonicity assumption on the autocorrelation function. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process and specifically is able to reproduce more realistic dependence structures. We also study the conditions of stationarity and positivity for the intensity and the strong mixing property for the increments. Furthermore we compute the likelihood, present a simulation method and discuss an estimation method based on the autocorrelation function. A simulation and estimation exercise highlights the main features of the CARMA(p,q)-Hawkes.

翻译：在本文中,我们引入了名为CARMA(p,q)-Hawkes进程的新模型,作为具有指数内核的霍克斯模型,这意味着自动关系函数和实验性证据的行为将严格减少,拒绝接受对自动关系函数的单一度假设。拟议模型是一个霍克斯进程,其强度遵循连续时间自动递减平均(CARMA)进程,具体来说能够复制更现实的依赖性结构。我们还研究强度和强力混合特性的静态和可视性条件。此外,我们计算可能性,提出模拟方法,并讨论基于自动关系函数的估计方法。模拟和估计活动突出CARMA(p,q)-Hawkes的主要特征。