A Hawkes model with CARMA(p,q) intensity

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.


Introduction
Point processes are useful mathematical models that describe the dynamics of observed event times and have been studied and applied in several fields of study from queueing theory to forestry statistics.Among the family of point processes the Hawkes (1971a,b) process is widely the most established and widespread model in different areas, especially in quantitative finance, actuarial science and seismology (see Ogata 1988 and references therein for further details).Indeed the Hawkes process is particularly interesting since it is a self-exciting process, which means that each arrival excites the intensity such that the probability of the next arrival is increased for some period after the jump, and consequently it is well-suited to investigate, for instance, natural clustering effects and bank default in time.To show the versatility of the Hawkes process we mention a arXiv:2208.02659v3[q-fin.ST] 20 Aug 2022 few other possible non-financial and non-insurance applications: a) social science area e.g., Mohler et al. (2011) for the modeling of urban crime and Boumezoued (2016) for the population dynamics; b) social media sector as done in Rizoiu et al. (2017); and c) the modeling of disease spreading such as COVID-19 transmission as discussed in Chiang et al. (2022).
Recently the Hawkes process has gained a relevant role in financial modeling, in particular in the field of market microstructure.As a matter of fact it is used to model market activity, especially order arrivals in the limit order book (e.g., Bacry et al., 2013;Muni Toke and Yoshida, 2017;Clinet and Yoshida, 2017).For a complete overview of applications of the Hawkes process in finance, we suggest the works of Bacry et al. (2015) and Hawkes (2018).The Hawkes process has aroused its appeal among researchers and practitioners as well as in the insurance area.Indeed, as mentioned in Lesage et al. (2022), insurance companies are interested in point processes for the quantification of regulatory capital and in managing risks (e.g., computing ruin probabilities and measuring the effect of cyber-attacks as discussed respectively in Cheng andSeol 2020 andBessy-Roland et al. 2021).Swishchuk et al. (2021) show that the use of a Hawkes process with exponential kernel for modeling insurance claim occurrences provides an improvement over the fit of a classical Poisson model.However, they are not able to fit different empirical autocorrelation functions as exhibited in Swishchuk et al. (2021, Figures 3 and 5, p. 112).
As stated in Errais et al. (2010), the Hawkes process with exponential kernel is Markovian and shows a good level of tractability that makes it useful for real applications in the presence of large data sets (e.g., high-frequency market data).The specification of the kernel restricts the shape of the time dependence structure of the number of jumps observed in intervals with same length.Indeed, as observed in Da Fonseca and Zaatour (2014), the autocorrelation in a Hawkes model is a decaying function of lags which is not flexible enough to represent the dependence structure observed in many data sets (e.g., wind speed data in which the exponential autocorrelation overshoots the empirical one for small lags and vice versa for large lags as documented in Benth and Rohde 2019;and, as shown in Hitaj et al. 2019, mortality rates where the empirical autocorrelation function of the shock term appears to be non-monotonic).
To overcome the aforementioned drawback, in this paper we introduce a new model named CARMA(p,q)-Hawkes process.The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process and it is able to provide several shapes of the autocorrelation function as it removes the monotonicity constraint detected in the standard Hawkes process.The greater flexibility relies on the CARMA(p,q) component of our model, especially in the choice of the autoregressive and moving average parameters.The CARMA process has been introduced in Doob (1944) and it is the continuous time version of the ARMA model.The advantage of the CARMA process, other than to design different shapes of autocorrelation functions, is to handle better irregular time series with respect to the ARMA process, especially for high-frequency market data, as discussed in Marquardt and Stelzer (2007) and Tómasson (2015).As a matter of fact, the CARMA model has found many applications in the literature.Here, we list a few of these applications: a) Andresen et al. (2014) use a CARMA (p,q) model for short and forward interest rates, while b) Hitaj et al. (2019) employ such a model in order to capture the dynamics of the shock term in mortality modeling; c) Benth et al. (2014) consider a non-Gaussian CARMA process for the dynamics of spot and derivative prices in electricity markets; and d) Mercuri et al. (2021) provide formulas for the futures term structure and options written on futures in the framework of a CARMA(p,q) model driven by a time-changed Brownian motion.As remarked in Iacus and Mercuri (2015), CARMA models have manifold interests: they can be used to describe directly the dynamics of time series and to construct the variance process in continuous time models (see Brockwell et al. 2006and Iacus et al. 2017, 2018 for further details).Our paper presents a different type of application as we use CARMA(p,q) models for the intensity of a point process.
In this paper, after reviewing the basic notions of the Hawkes and the CARMA processes, we introduce the CARMA(p,q)-Hawkes process and study the conditions of stationarity and positivity for the intensity, the autocorrelation function of the process and prove the strong mixing property of increments that leads us to the asymptotic distribution of the empirical autocorrelation function.
The remainder of the paper is organized as follows.Section 2 reviews the Hawkes process with exponential kernel while Section 3 presents the CARMA(p,q) model in the Lévy setting.Section 4 introduces the CARMA(p,q)-Hawkes process.Section 5 focuses on the autocorrelation function of the jumps in the proposed model and its asymptotic distribution, while Section 6 presents a simulation and an estimation exercise.Section 7 concludes the paper.

The Hawkes Process
Point processes are useful to describe the dynamics of observed event times, i.e., a collection of realizations {t i } ∞ i=1 , t i ≥ 0 for i = 1, 2, . . .with t 0 := 0 of the non-decreasing non-negative process {T i } i≥1 called the time arrival process.The counting process N t , representing the number of events up to time t, is obtained from the time arrival process as follows: for t ≥ 0 with associated filtration (F t ) t≥0 that contains the information of the counting process N t up to time t.An important quantity when dealing with a point process N t is the conditional intensity λ t defined as: It then follows that the counting process satisfies the following properties The conditional intensity λ t of a general self-exciting process has the following form: with baseline intensity parameter µ > 0 and (excitation) kernel function h (t) : [0, +∞) → [0, +∞) that represents the contribution to the intensity at time t that is made by an event occurred at a previous time T i < t.Following the general results about the Hawkes process in Brémaud and Massoulié (1996), the stationary condition reads: The most used kernel is the exponential function and specifically h (t) = αe −βt with α, β ≥ 0. The stationary condition in (3) implies β > α while to prove the Markovianity of the couple (λ t , N t ) it is enough to rewrite the intensity for any s < t as Observing that From (4) we have that the distribution of the intensity λ t given the information at time s depends only upon λ s and on the increments of the counting process over the interval [s, t), which depend on the conditional intensity itself implying that the couple (λ t , N t ) is itself Markovian.The intensity λ t is the solution of the following differential equation: Exploiting the Markovianity of the process X t := (λ t , N t ), it is possible to get the infinitesimal generator (see Errais et al. 2010 andDa Fonseca andZaatour 2014 for further details) associated to a function f : R + × N → R with continuous partial derivatives with respect to the first argument ∂f ∂λ (x).Starting from the definition of the infinitesimal operator for a Markov process X t , that is: Errais et al. (2010) compute the infinitesimal generator for the Hawkes process with exponential kernel that reads For every function f in the domain of the infinitesimal generator it is possible to build a martingale process M t with respect to the natural filtration in the following way: that leads to the well-known Dynkin's formula The above formula for f ≡ N t has been used in Da Fonseca and Zaatour (2014) to compute the moments and the autocovariance function of jump increments observed in intervals of length τ with lag δ. 1.The long-run expected value of the number of jumps during an interval of length τ is 2. The long-run variance of the increments reads 3. The long-run covariance of the number of arrivals for two non-overlapping intervals of length 4. The long-run autocorrelation function of the number of jumps over intervals of length τ separated by a time lag of δ reads and is always positive for α < β (stationarity condition) and exponentially decaying with the lag δ.
From ( 8) and ( 9) it is clear that the Hawkes model with exponential kernel can reproduce only strictly decreasing autocorrelation functions for varying lag values δ.An interesting extension is given in Boswijk et al. (2018) where self-excitation is identified through the modeling of common jumps between the log price process and its own jump intensity.

Lévy CARMA(p,q) models
The formal definition of a Lévy CARMA(p,q) model Y t with p > q ≥ 0 is based on the continuous version of the state-space representation of an autoregressive moving average ARMA(p,q) model.In particular we have that where X t satisfies the following stochastic differential equation and {Z t } t≥0 is a Lévy process.The p × p matrix A has the following form with b q+1 = . . .= b p−1 = 0. Given a starting value for X s , the solution of ( 11) is Brockwell et al. (2011, Section 2, p. 251), under the assumption that ensures the stationarity of X t (i.e., all eigenvalues λ1 , . . ., λp of matrix A are distinct with negative real part 1 ), the CARMA(p,q) model can be written as a summation of a finite number of continuous autoregressive models of order 1, which are also known as CAR(1) models.Specifically, a (1) (z) and the polynomials a (z) and b (z) are defined as Note that a (1) (z) is the first derivative of the polynomial a(z).
Remark 1.The eigenvalues of matrix A denoted by λ1 , . . ., λp are the same as the zeros of the autoregressive polynomial a(z).As observed in Tsai and Chan (2005), the assumption that the zeros of a(z) have negative real parts is a necessary condition for the stationarity of the CARMA(p,q) process Y t .
Definition 1.A stationary CARMA(p,q) process Y t where Z is a second-order subordinator can be equivalently defined as: where the function h(t) = b e A 1 [0,+∞) (t)e is the kernel of the CARMA(p,q) process.As Y t is independent of Z s − Z t , ∀s ≥ t, the process Y t is said to be a casual function of the subordinator Z t , also known as casual CARMA(p,q) model.
In Brockwell and Marquardt (2005) it is shown that the function h(u) can be written as 1 The eigenvalues are sorted based on their real part in an increasing order.
The positivity conditions for the kernel and for the process itself, which is for instance required for modeling the volatility using CARMA(p,q) models, have been deeply investigated in Tsai and Chan (2005) and in Benth and Rohde (2019) for the case of positive subordinators.
4. CARMA(p,q)-Hawkes model In this section, we introduce a point process where the intensity follows a CARMA(p,q) process that is a generalization of the Hawkes process with an exponential kernel.
4.1.CARMA(p,q)-Hawkes: stationarity and positivity conditions for the intensity Definition 2. A vector process [X 1,t , . . ., X 1,p , N t ] of dimension p + 1 is a CARMA(p,q)-Hawkes process if the conditional intensity λ t of the counting process N t is a CARMA(p,q) process driven by N t and has the following form: in which the baseline parameter µ is strictly positive and the vector b is defined as in (14).The vector where the companion matrix A and the vector e have respectively the form in (12) and (13).
The dynamics of the state space process X t in ( 18) is described through a linear stochastic differential equation and it is a Markov process.Consequently, a CARMA(p,q)-Hawkes process is in turn a Markov process.
Remark 2. The stochastic differential equation in (18) has an analytical solution given the initial condition, that is The non-decreasing and non-negative trajectories of the counting process N t imply the positiveness of λ t for non-negative kernel functions.
To investigate the stationary regime of a CARMA(p,q)-Hawkes model, it is necessary to determine the conditions required for a non-negative kernel, i.e., b e At e ≥ 0, ∀t ≥ 0. In case of a CARMA(p,q) driven by a non-negative Lévy process the conditions of a non-negative kernel are presented in Tsai and Chan (2005, Theorem 1, p. 592).In a similar fashion such conditions can be applied directly to our case due to the non-negative trajectories of the counting process N t .
Indeed, as done in Brockwell et al. (2006, Theorem 5.2) for COGARCH(p,q) models, we rephrase their results for a generic CARMA(p,q)-Hawkes process when b 0 > 0 in the next proposition.
(a) For a CARMA(p,q)-Hawkes process such that the real part of all eigenvalues of A is negative, the kernel function h (t) := b e At e1 {t≥0} is non-negative if and only if the ratio function is completely monotone 2 on (0, +∞).
(b) A sufficient condition for the kernel function of a CAR(p)-Hawkes process to be non-negative is that all eigenvalues of A are real and negative.
(c) A sufficient condition for the kernel function of a CAR(p)-Hawkes process to be non-negative is that if λi 1 , λi 1 +1 , . . ., λir , λir+1 is a partition of the set of all pairs of complex conjugate eigenvalues of A (counted with multiplicity), then there exists an injective mapping u : {1, . . ., r} → {1, . . ., p} such that λu(j) real eigenvalue of A satisfies λu(j) ≥ Re λi j .
(d) For a non-negative kernel h (t) in a CAR(p)-Hawkes process, it is necessary to find a real eigenvalue λi such that λi ≥ Re λj where j = 1, . . ., p with j = i.
(e) Suppose all eigenvalues of A are negative real numbers sorted as follows λp ≤, . . ., ≤ λ1 and that all the roots of b (z) = 0 are negative real numbers such that γ q ≤, . . ., then the kernel of a CARMA(p,q)-Hawkes process is non-negative.
We remark that the non-negativity requirement for the kernel implies a strictly positive intensity process λ t as the baseline parameter µ is strictly positive.
Without loss of generality, we assume that matrix A is diagonalizable which corresponds to the assumption that the eigenvalues of A are distinct.The eigenvectors of A are 1, λj , . . ., λp−1 , j = 1, . . ., p used to define a p × p matrix S as .
2 A function f (x) defined on (0, +∞) is said to be completely monotone if and only it has derivatives of all orders and (−1 It follows that S satisfies S −1 AS = diag λ1 , . . ., λp , a result used to prove the next proposition on the stationarity conditions for a CARMA(p,q)-Hawkes process.
Proposition 3. Let us consider a non-negative kernel function and suppose µ > 0. Then a CARMA(p,q)-Hawkes (X 1,t , . . ., X p,t , N t ) is a stationary process if all eigenvalues of A are distinct with non-negative real part and −b A −1 e < 1.
Proof.For a non-negative kernel function, the stationary condition in (3) for a CARMA(p,q)-Hawkes process becomes where I is the identity matrix with dimension p.As A is diagonalizable, where Λ := diag λ1 , . . ., λp .Thus the limit in ( 20) is lim Recalling that all eigenvalues of A have negative real part, we notice that e ΛT tends to a p × p zero matrix.The integral in (20) becomes The stationarity condition in (3) implies −b A −1 e < 1.
Assumption 1.We shall assume for the remainder of the paper that: i) the kernel is a nonnegative function and µ > 0; and ii) all eigenvalues of A are distinct with negative real part and b A −1 e > −1.
For practical applications, instead of checking ex-post signs of eigenvalues of matrix A, it is possible to enforce ex-ante the negativity of the real part for eigenvalues using some transformations on the parameters space as done, for example, in Tómasson (2015).As a CARMA(p,q)-Hawkes process is Markovian, we are able to calculate the infinitesimal operator as described in the following proposition.
Proposition 4. Let f : R p × N → R be a function with continuous partial derivatives with respect to the first p arguments.Under the same conditions in Assumption 1, the infinitesimal generator of function f for a CARMA(p,q)-Hawkes process is: where A [p,] is the p-th row of the companion matrix A and the intensity process λ t is defined as in (17).Alternatively, the infinitesimal generator can be written as where T +h where X NJ T +h means no jump (NJ) occurred in the interval (T, T + h] and can be written in the following way as the quantity T +h T e A(T +h−t) edN t is zero due to the absence of jumps in the interval (T, T + h]. From If N T +h − N T = 1 then X T +h := X 1J T +h is computed as Defining the jump time T h in the time interval (T, T + h] we get

As lim h→0
T h = T , we observe that lim h→0 Note that X t + e = [X t,1 , . . ., X t,p + 1] and consider the following quantity: The infinitesimal generator is: From ( 24) and ( 25) we obtain To compute the limit (26) we use De l'Hôpital's rule and substituting ( 27) in ( 26), we finally obtain the result in (23).
The conditional expected value for f (X 1,T , . . ., X p,T , N T ) can be computed applying the Dynkin's formula: that has a representation of the following form with initial condition f (X 1,t 0 , . . ., X p,t 0 , N t 0 ).We use the infinitesimal generator (23) and the result in (28) to obtain the following proposition for the computation of the first moment of the counting process N t .In the remainder of the paper, we use Proposition 5. Let Ã be a p × p companion matrix where the last row has the following structure Under Assumption 1 and supposing that all eigenvalues of Ã are distinct with negative real part, for any T > t 0 ≥ 0, the conditional first moment of the counting process is while the conditional expected value of the state process X T is The quantities in (31) and (32) satisfy respectively the following ordinary differential equations: with initial conditions3 X t 0 and N t 0 .The long-run value for Moreover, the expected number of events that occurs in an interval with length τ , i.e., (T, T + τ ], given the information at time t 0 < T is and the stationary behaviour of (36) is Proof.To determine the expected number of jumps in (31) we obtain first the infinitesimal generator of the function f (X 1,t , . . ., X p,t , N t ) = N t , that is Af t = λ t where the conditional intensity λ t is defined in (18).Applying the Dynkin's formula in (29) we obtain the following ODE Then, we compute E t 0 [X t ] that requires a system of infinitesimal generators.In particular, for i = 1, . . ., p − 1, we have and Applying (29), we get where Ã is defined in (30).With the initial condition X t 0 , the solution of the system in ( 39) is (32).Substituting ( 32) in ( 38) we obtain the following ODE for the expected number of jumps whose solution is in (31) with initial condition N t 0 .Using the result in (31) we observe by straightforward calculations that the expected number of jumps in an interval of length τ reads as in (36).
Due to the negativity assumption for the real part of the eigenvalues of matrix Ã, we obtain the asymptotic behaviour in ( 35) and ( 37) as lim T →+∞ e ÃT = 0 where 0 is a p × p zero matrix (see A sufficient condition for (40) is where b 2 := p i=1 b 2 i−1 is the Euclidean norm of b, σ max (S) and σ min (S) are maximal and minimal singular values of S. In particular, we observe that and that k 2 (S) := S 2 S −1 2 , the condition number in 2-norm, can be written as Moreover, denoting with eb F the Frobenius norm of eb , we obtain eb 2 ≤ eb F .

Simulation and Likelihood Estimation of the CARMA(p,q)-Hawkes
We propose a simulation method for the CARMA(p,q)-Hawkes model following the same idea presented in Ozaki (1979, Section 4, p. 148).
Suppose that T 1 , . . ., T k , which correspond to time arrivals, are already observed.Then it is possible to simulate the next time arrival T k+1 by generating a random number from a standard uniform distribution, i.e., U ∼ Unif (0, 1), and by solving this equation with respect to u: The conditional intensity λ t can be replaced by ( 17) and X t can be substituted by ( 19) obtaining so ln Developing and rearranging the right-hand side of (47) we have where the integral in the second equality is computed using the results in (A.1).Defining S (k) : The quantity S (k) can be obtained recursively as follows Note that a similar recursive expression has been obtained in Ozaki (1979) for a Hawkes process with exponential kernel.
As follows we present the likelihood of a CARMA(p,q)-Hawkes model.Consider that θ = (b 0 , . . ., b q , a 1 , . . ., a p ), then the likelihood of a CARMA(p,q)-Hawkes model is given by Exploiting the fact that can be written as and recalling once again that X t can be expressed by ( 19) and rearranging the expression we have Working on the inner integral, the likelihood becomes while using the results in (A.1) we get Developing the integral in (53) and recalling that S (k) := k i=1 e A(T k −T i ) , we finally obtain that the likelihood of a CARMA(p,q)-Hawkes model writes 5. Autocovariance and Autocorrelation of a CARMA(p,q)-Hawkes process In this section we compute the stationary autocorrelation and autocovariance functions for the number of jumps in non-overlapping time intervals of length τ .To this aim we introduce some quantities that are useful to compute the asymptotic covariance of a CARMA(p,q)-Hawkes process.
Here an example of the matrix Ã for a CARMA(3,2)-Hawkes model The second quantity introduced is the p × p(p+1) 2 matrix B defined as: The third quantity is the p(p+1) 2 × p matrix C in which the entry in the i− th row and in j− th column has the following structure Let H be a p × 1 vector.

Conditions for existence of stationary autocovariance function
We rewrite the quantity E t 0 X T X T b using the vlt (•) operator defined in (58).
Lemma 1.The following identity holds true where the matrix B is defined in (56) and the operator vlt (•) is defined as in (58).Moreover: Proof.Using the definition of matrix B in (56), the identity in ( 59) is straightforward.To show the result in ( 60), we need first to compute the infinitesimal generator for each component of vlt X t X t .From the definition in (58) we identify p blocks where the dimension of each block decreases by one unit.More precisely, the j−th block has p − j + 1 elements.Considering the first block (i.e., j = 1) we have p infinitesimal generators obtained applying the result in ( 22) of Proposition 4. For the first element in the first block, we have AX 2 t,1 = 2X t,2 X t,1 .While for the i−th element in the first block with i = 2, . . ., p − 1 we get AX t,i X t,1 = X t,i X t,2 + X t,i+1 X t,1 and finally For a generic j−th block, we get p − j + 1 infinitesimal generators.In particular for i = j we have The last block contains only one infinitesimal generator of the form Using the Dynkin's formula in ( 29) we obtain the following system of linear ODE's: where the p(p+1) vector ẽ is composed of zero entries except the last position where the element is one; Ã and C are defined in ( 55) and ( 57) respectively.
The first step is to solve the ODE defined in ( 61) whose solution has the following form We also observe that Substituting ( 63) into ( 62) we obtain the result in (60).
Appendix D contains the proofs of the following two propositions on the variance and covariance of the number of jumps that occur in two non-overlapping time intervals of the same length for a CARMA(p,q)-Hawkes model.
Proposition 6.Under Assumption 1 and supposing that all eigenvalues of Ã and Ã have negative real parts, the long-run covariance Cov (τ, δ) defined as in (8) for a CARMA(p,q)-Hawkes process has the following form: where g ∞ (τ ) is defined as Proposition 7.Under the same assumptions as in Proposition 6, the long-run variance V ar (τ ) of the number of jumps in a interval with length τ for a CARMA(p,q)-Hawkes process, defined as in (8), has the following form: where h ∞ (0) is defined as Remark 4. Combining the results in Propositions 6 and 7, we determine the asymptotic autocorrelation function of number of jumps in non-overlapping time intervals of length τ , i.e., ρ τ (d), for a CARMA(p,q)-Hawkes in a closed-form formula: where d denotes the lag order.

Strong mixing property for the increments of a CARMA(p,q)-Hawkes and asymptotic distribution of the autocorrelation function
The asymptotic distribution of the autocorrelation function of a CARMA(p,q)-Hawkes process can be easily obtained if we show that the increments of the process are strongly mixing.
Definition 3. Let (Ω, F, P) be a probability space and A, B two sub σ−algebras of F. The strongmixing coefficient is defined as: Following Poinas et al. (2019), the quantity in ( 69) can be reformulated for a point process N t in the following way: where ξ b a denotes the σ−algebra generated by the cylinder sets on the interval (a, b] 4 .Considering the sequence (∆ 1 N k ) k∈Z where ∆ 1 N k := N k+1 −N k is the number of jumps in the interval of length 1 and extremes k, k + 1, the strong-mixing coefficient has the form where F b a is the σ−algebra generated by the sequence (∆ 1 N k ) a≤k≤b .If α N (r) → 0 (respectively α ∆ 1 N k (r) → 0) as r → +∞, the point process N t (respectively ∆ 1 N k ) is said to be strongly-mixing.
Using Theorem 1 in Cheysson and Lang (2020), we obtain the following proposition.
Proposition 8.A CARMA(p,q)-Hawkes process satisfying Assumption 1 is strongly mixing with exponential rate.
Proof.We first prove the existence of a positive constant a 0 > 0 such that the kernel function We notice that Assumption 1 implies that Choosing a 0 ∈ (0, |Re (λ 1 )|) the condition in ( 72) is ensured and thus we can apply the result in Theorem 1 proved by Cheysson and Lang (2020), and the strong-mixing coefficient results to be α N (r) = O (e −ar ) where a ∈ (0, a 0 ).
As shown in Cheysson and Lang (2020), we have that α ∆ 1 N (r) ≤ α N (r) and the result in Proposition 8 implies that the sequence (∆ 1 N k ) k∈Z is strongly mixing.This result is useful to determine the asymptotic distribution of the sample autocovariance and autocorrelation functions associated to the sequence (∆ 1 N k ) k∈Z .Following the result in Ibragimov and Linnik (1971), we obtain the following result for the asymptotic distribution of the sample mean, the sample variance and the sample autocovariance function.
Proposition 9. Let (N t ) t≥0 be a stationary CARMA(p,q)-Hawkes process that satisfies the assumptions in Proposition 8. We assume the existence of a positive constant φ such that E (∆ 1 N 1 ) 4+φ < of locally finite counting measures on Ω.Then the σ−algebra ξ b a is defined as: +∞. Denoting with , with k = 1, ..., n and d < n as n → +∞, we have: Acv (1) . . .
where Acv (d) := Cov(1, d − 1) and Proof.The proof is quite standard and is an application of Theorem 18.5.3 in Ibragimov and Linnik (1971) and Cramér-Wold device.Denoting with we apply Theorem 18.5.3 in Ibragimov and Linnik (1971) to the linear combination c V k k=1,2,...n where c is a generic d + 2 real vector such that c Σc > 0. Since the strong mixing property is preserved under linear transformations as well as the rate we have Applying Cramér-Wold device we obtain the asymptotic behavior in (73).

Conclusion
In this paper we introduce a new Hawkes process where the intensity is a CARMA(p,q) model.
We analyze the statistical properties of this process and obtain a closed-form expression for the autocorrelation function of the number of jumps observed in non-overlapping time intervals of same length.The model is a generalization of the standard Hawkes with exponential kernel but it is able to reproduce more complex dependence structures similar to those documented in several data sets.We provide below the proof of Propositon 6 on the long-run covariance of the number of jumps in a CARMA(p,q)-Hawkes model.
Proof.We first determine the covariance of number of jumps in two non-overlapping time intervals given the information at time t 0 .This quantity is formally defined as Using the iteration property of the conditional expected value, (D.1) becomes Applying the result (36) in Proposition 5, we get where In the rhs of (D.3), the last two terms are stationary due to the result in (37) and to the negativity of the real part for the eigenvalues of Ã; the third term converges to zero as t → +∞ while the fourth term has the following limit behaviour For the first two terms in the rhs (D.3) consider the quantity: as t → +∞.In (D.5) the vector E t 0 [N t X t ] requires the calculation of p infinitesimal generators.
We then observe that for i = 1, . . ., p − 1, the infinitesimal generator of the function N t X t,i is: while for i = p from where we get The quantity E t 0 [X T N T ] is not stationary but it is useful as it appears in the rhs of the function h t 0 (t, τ ) introduced in (D.5) that can be rewritten as We analyze the long-run behaviour of each term in the rhs of (D.8).We first observe that The fourth term in the rhs of (D.8) can be written as Integrating by parts we get Using the formula for the conditional expected value of the counting process in (31) we get The fifth term in the right-hand side of (D.8) can be rewritten as that has the following long-run behaviour lim Lemma 1 suggests that the last term in the rhs of (D.8) can be written as The result in (A.1) implies that To determine the asymptotic behaviour of this term, we analyze the long-run behaviour of the Here we provide the proof of Proposition 7.
Proof.For the asymptotic variance we need to compute the conditional variance of the number of jumps in an interval with length τ .First we observe that We then compute the second moment of the increments .
For E t 0 N 2 t it is useful to compute the infinitesimal operator for the function f (X 1,t , . . ., X p,t , N t , ) = N 2 t , that reads Af t = µ (2N t + 1) + 2b N t X t + b X t .

Remark 3 .
The result in (37) becomes (6) if we consider a CAR(1)-Hawkes with b 0 = α and a 1 = β.Using the same arguments inBrockwell et al. (2006, proof of Proposition 4.1, p. 815) , all eigenvalues of matrix Ã have negative real parts if for some positive integer r ≥ 1 this context, • r denotes the natural matrix norm induced by the vector L r -norm.This result comes directly from an application of the Bauer-Fike Theorem (seeBauer and Fike 1960 for further details) since Ã is obtained by perturbing matrix A as Ã = A + eb .
i-th row is the result of a row concatenation of p vectors with dimensions p, p − 1, . .., p − i, . . . 1, respectively.The first i − 1 vectors have zero entries except the element in position i that coincides with b i−1 , the vector with dimension p − i contains the elements b i , . . ., b p−i and the remaining vectors have zero entries.For example, in the case of a CARMA(3,2)-Hawkes model, the structure of matrix B reads

R e a 0
|t| h (t) dt = b +∞ 0 e a 0 t e At dte = b S +∞ 0 e a 0 t e Λt dtS −1 e.

Figure 3
Figure 3(a) exhibits a simulated trajectory of the counting process N t , while Figure 3(b) displays the ACF at different lags.Figure 3(b) is a clear example of the fact that a CARMA(3,1)-Hawkes process can accommodate different shapes of autocorrelation structures rather than an exponential monotonic decay behaviour as it is the case of a standard Hawkes process.
Proofs of propositions on long-run covariance and variance of the number of jumps

Table
Table1exhibits the MLE estimates for the parameters and the number of occurred events for different levels of final time T , whereas Table2shows the estimated parameters using the MME

Table 3 :
Parameter estimates from MLE and number of occurred events NT for a CARMA(3,1)-Hawkes process.