In this paper we introduce a new model, named CARMA(p,q)-Hawkes, as the Hawkes model with exponential kernel implies a strictly decreasing behavior of the autocorrelation function while empirical evidences reject its monotonicity. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process. We also study the conditions for the stationarity and the positivity of the intensity and the strong mixing property for the increments. Furthermore, we present two estimation case studies based respectively on the likelihood and on the autocorrelation function.

中文翻译：

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具有 CARMA(p,q) 强度的 Hawkes 模型

在本文中，我们引入了一种新模型，称为 CARMA(p,q)-Hawkes，因为具有指数核的 Hawkes 模型意味着自相关函数的严格递减行为，而经验证据拒绝其单调性。所提出的模型是霍克斯过程，其中强度遵循连续时间自回归移动平均 (CARMA) 过程。我们还研究了增量的平稳性和强度正性以及强混合特性的条件。此外，我们提出了两个分别基于似然函数和自相关函数的估计案例研究。