Measuring Discrepancies Between Poisson and Exponential Hawkes Processes

Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process, whose properties are therefore analysed before introducing the measures. Such quantities can be also used to detect mistakes in parameters estimation.


Introduction
A Poisson process models, by means of the unordered vector of its jump times, the occurrence of similar and independent events. This means that the jumps of a Poisson process are in a sense unexpected and the Poisson process, despite its mathematical tractability, is an inadequate tool when we believe that some connection exists among events; in particular when some events are caused by some previous events and it becomes therefore possible to predict or even to prevent them.
A widespread tool for modelling this kind of phenomenon is through the use of Hawkes processes.
The name Hawkes processes is due to the seminal work (Hawkes 1971), setting the theoretical basis for the study of self-exciting processes (see also Daley and Vere-Jones 2008;Bacry et al. 2015 and references therein), that were actually already used in engineering and reliability theory (see e.g. Rangan and Grace 1988 and references therein). Currently Hawkes processes are applied in a number of fields: in geology, to earthquakes or volcanic eruptions, in biology, to population growth, spread of infections, neuronal activity, in computer science or social sciences, to networks and social interactions, and in finance, to order book dynamics, defaults and so on (see e.g. Zhuang et al. 2002;Reynaud-Bouret et al. 2013;Delattre et al. 2016;Hawkes 2018).
The main feature of this class of processes is that they exhibit self-excitation behaviour, that is the jump intensity increases immediately after each jump. It is reasonable to require that such a self-excitation effect is limited in time and decays as time elapses. It is also reasonable and very frequent in the above applications, assuming the decay to have an exponential form. This features correspond to a specific expression of the stochastic intensity in the exponential Hawkes model, that will be our main object of investigation: where t ∈ R + denotes time, T i the random time of the i-th jump and λ 0 , α, β are deterministic, constant and positive parameters. The exponential Hawkes model also benefits of a particular mathematical tractability, mainly due to its Markov property.
We are interested in quantitatively describing this stochastic intensity and further characteristics, with the aim of distinguishing between Hawkes and Poisson processes. Towards achieving this aim, we present and study a variety of measures, quantifying the shift between the two processes.
In general, there are three possible scenarios, that would have to be separately discussed. We may deal with measuring: -discrepancy between a Hawkes process and a Poisson one; -discrepancy between a Hawkes process and a process not having a specific or a known law; -discrepancy between a Poisson process and a process not having a specific or a known law.

Notation and Basic Results
Definition 1 A simple counting process N = {N t } t≥0 with natural filtration F = σ (N s , 0 ≤ s ≤ t) is a Hawkes process if it has a stochastic intensityλ given bȳ where λ 0 ∈ [0, +∞) is constant, and the kernel Φ ≥ 0 is a deterministic function, locally integrable on R + , wrt the Lebesgue measure.
Conventionally, we set N 0 = 0, while we denote by {T n } n∈N the random jump times of N .
In particular, we consider an exponential decay of self-excitation. Hence the intensity can be written asλ t = λ 0 + α T i <t e −β(t−T i ) ; note thatλ turns out to be a càglàd process.
Remark 1 In order to deal with a process with a finite number of jumps on limited intervals, we set α < β. Otherwise, the excitation effect would overcome the decay and the process would be explosive. For processes on R, the condition α < β is equivalent to stationarity. However, when the process starts at λ 0 from t = 0, then α < β implies asymptotic stationarity.
Given F t− , we know exactly when the jumps before t occurred and, to indicate that they are no longer random, we name them t n rather than T n . Given F t− , we are able to reconstruct the path realized by the intensity up to t−, and we call λ t the intensity process conditional to F t− . In particular λ t is such that i.e. and We recall some formulas that will be used or generalized in the following. For notational simplicity, since we will deal with absolutely continuous distributions, we will write the probability densities as P (T = t) instead of P (T ∈ [t, t + dt)). Reference (Rangan and Grace 1988) provides an expression for the conditional density By recursively applying (3), the finite-dimensional distribution of the process can be computed as and, by changing variables, the distribution of the first n inter-arrival times, In parallel with the intensity (conditional or stochastic), a cumulative or integrated intensity is defined as Daley and Vere-Jones (2008), Karr (1991), and Rasmussen (2011). We recall two relevant properties of (t): can be seen as the expected number of jumps of the process N on [0, t].
In case of stationarity, are the inter-arrival times of a standard Poisson process.

Analytical and Distributional Properties of Stochastic Intensity and of the Integrated Intensity
As mentioned above, the first important difference between a (homogeneous) Poisson process and a Hawkes process concerns the characterization in terms of the jump intensity: a constant for the first one, a collection of random variables with distribution depending on the process itself, in the second case.
The intensity of a Hawkes process has also a specific form, highlighting the main difference of behaviour of such a process with respect to a Poisson one, that is self-excitation, i.e. the occurrence of a jump increases the probability of further jumps.
In the literature, in particular in the reliability field, the intensity also has a key role in describing dynamic dependence properties among events.
Therefore we believe that those aspects of diversity of a process from a Poisson one, that concern dependence among jumps, may be well described and measured at first by quantities derived from the intensity.
We start this section by studying some properties of the distribution of the intensity of a Hawkes process, in particular of the distribution of the intensity just before jumps.
First of all, we recall a recursive formula for λ t : let F t− be given and, in particular, t n be the time of the last jump before t. Then (see Foschi et al. 2019).
Remark 2 We notice that applying (4) does not require the knowledge of F t− . For computing λ t , it is sufficient to know the value of λ t n and that t n is the time of the last jump before t. Therefore, given λ t n , the process after t n is independent of the history of the process before t n

Intensity Values Just Before a Jump Time
Let T n denote the random time of the n-th jump,λ T n the intensity immediately before the nth jump and E n t the event {no jumps in (T n , T n + t)} and letλ T n be given. Then an analogue of Eq. 4 holds for the stochastic intensity as well: Remark 3 Equation 5 can be generalized to computeλ τ +t|E , with τ a given time instant, not necessarily a jump time, and E a suitable event (see Foschi et al. 2019).

that is
Lemma 1 The process {λ T n } n∈N is markovian.
By Lemma 1, we can prove Theorem 1 The following recursive formula holds: Proof As a first step, we compute the distribution ofλ T 2 , partitioning wrt the variable T 1 , i.e.

P (λ T n
where the integration limits are set in view of Foschi et al. (2019), Proposition 1.
that, by Foschi et al. (2019), Proposition 2, and by some algebra, can be written as By substituting in Eq. 8, the thesis is proven.
Remark 4 Notice that, by definition,λ T 1 = λ T 1 is the intensity immediately before the first jump and therefore it coincides with λ 0 , i.e.
The thesis follows by Remark 4 and Eq. 9.
Remark 5 This result is particularly useful and of straight application: in fact, recursively writing the distribution ofλ T n allows us to easily implement it. Furthermore, e.g. in Reliability, the functionλ t can be interpreted as an analogue of the instantaneous wear of a system (see e.g. Cha and Finkelstein 2012), causing the failure of that system when it exceeds a certain (deterministic) threshold. By applying (6) or (10), we are now able to compute the probability of reaching such a threshold. Similarly the (stochastic) integrated intensity¯ (t) can be used to represent the cumulative wear of a system up to time t and an analogue of Eqs. 6 or 10 can be applied to it.
Step-Wise Convexity of λ t For any t ∈ R \ {t 1 , t 2 , . . . }, λ t is infinitely many times differentiable. In particular, that is λ t is decreasing in the intervals between two jumps and dλ t dt is negative but increasing, meaning that it is decreasing in absolute value.
In a neighbourhood of two consecutive jumps times, we have dt , that implies that, after the jump, the slope is still negative but more pronounced than before.

Jumps' Concentration Impact on Intensity and Integrated Intensity
Unlike for Poisson and mixed Poisson processes, for Hawkes processes, the observation of the exact configuration of jumps (e.g. till time t) has a greater information content than the observation of the number of jumps occurred till time t.
This fact and the non-homogeneous concentration of jumps are traits distinguishing a Hawkes process from a Poisson one.
In this subsection, we aim at showing how a different position or a non-homogeneous concentration may affect the intensity and the integrated intensity.
Let us fix δ > 0, k, m ∈ N with m > k, and define λ (k) (n+m)δ as the intensity at time (n + m)δ conditional on the fact that on [nδ, (n + m)δ) a jump was registered on k different intervals of length δ, and on m − k intervals there were no jumps. Given k, a bigger value of m would indicate that the k jumps can possibly be less concentrated, while for given m a higher value of k means that on the same time period [nδ, (n + m)δ) more jumps occurred. One would expect that, given k, λ (k) (n+m)δ is non-increasing in m, because the impact of each jump had more time to decay, while, for a given m, λ (k) (n+m)δ is increasing in k. However this last fact turns out not to be necessarily true, because it depends on how the k intervals, where the jumps occurred, are distributed among the m ones.
To prove this fact, we split k as k + k , where k is the number of consecutive jumps occurring on the time interval [(n + m − k )δ, (n + m)δ), i.e. just before the observation time (n+m)δ, and k is the number of jumps distributed in some way on the remaining n−k −1 intervals. Note that, by construction, we have no jumps on [(n+m−k −1)δ, (n+m−k )δ).
We then show that, when k increases in such a way that k decreases and k increases, we can find configurations such that λ (k) (n+m)δ is not increasing. In fact, for a given m, we make an increase of k by some amount h.
where h is the decrease in the number of recent consecutive jumps, so that k passes to k − h , and h is the increment of k . We have to account for the following constraints: We point out that we represent by k a same number of jumps, but not the same jumps for both intensities. The term λ (n+m)δ may be negative and its minimum value is attained when the h jumps are the last ones of the k ones and the h are positioned starting from the k − h + 2-th place.
Then a sufficient condition guaranteeing that λ (k) (n+m)δ be increasing in k, regardless of the allocation (k , k ), is While the intensity value may be affected by the jumps configuration even in opposition to a greater number of jumps, as concerns the integrated intensity, we choose to analyse the impact on of a different concentration of the same number of jumps on a fixed interval I = [a, b).
Proposition 1 Let k ≥ 2 be the number of jumps occurred in a fixed interval I . The sparser the configuration of such jumps, the smaller the generated (I ) ≡ (a, b).
Proof We consider the value ofλ a is given and distinguish two cases: In view of the arbitrariness of I , the case "k jumps in [b − δ, b)" is analogous.
We compute the cumulative intensity in the two cases, respectively eq (I ) and δ (I ).
Equidistant jumps on I means that the jumps have occurred at times a + j b−a k , for j = 0, ..., k − 1.
In view of Eq. 4, this fact allows us to write The integrand can be simplified as follows: By integrating wrt τ on 0, b−a k , we obtain Therefore, we can compute eq (I ) as finally obtaining . (11) Let us split δ (I ) = eq (a, a + δ) + decay (a + δ, b), by supposing that the k jumps are equidistant on the interval [a, a + δ), so that we can apply (11) to compute eq (a, a + δ), while decay is the integrated intensity on the interval [a + δ, b) without jumps.
By recursively apply (4) (see also Foschi et al. 2019), By summing up the two terms, we obtain At first, we want to check that, for any δ ∈ (0, b − a), δ (I ) ≥ eq (I ). At this aim, we compute .

Non-poissonianity Measures
This section is devoted to the definition of three kinds of measures and to illustrating their specific properties and aims. Our measures naturally apply to single paths, that we may observe in real situations. However, they can be computed as well on samples of simulated paths of a process. This allows us to quantify how much a Hawkes process with given parameters is far from a Poisson process (even if no data are available) and also provides us with confidence intervals for the different measures, that are useful to possibly compare the values of measures computed on a sample with the theoretical ones. We introduce the following notation: Definition 2 Given λ 0 , α, β > 0, with α < β, we denote by N (λ 0 ,α,β) the exponential Hawkes process with parameters (λ 0 , α, β) and by H (λ 0 ,α,β) = H

Intensity-Based Measure
By definition, the intensity of a Poisson process is a deterministic constant λ 0 , i.e. a degenerate random variable, that therefore has a standard deviation equal to zero. As we illustrated above, for a Hawkes process N (λ 0 ,α,β) , both {λ t } t∈R + and {λ T n } n∈N are stochastic processes.
We aim at quantifying how much the intensity values are dispersed with respect to a constant. In order to do that, we consider the frequency distribution of {λ T n } n∈N . To obtain the values of {λ T n } n∈N , we need to know (or to estimate) the parameters of the Hawkes process; we can alternatively directly estimate the empirical intensity from data with any known method.
Since the intensity of a Poisson process should attain only one value, under this aspect, the distance of a Hawkes process from a Poisson process can be expressed by the dispersion M disp of the values attained byλ T n for n ∈ N. Namely Definition 3 Given a realization H of the Hawkes process N (λ 0 ,α,β) , let j be the number of its jumps and n denote, for n = 1, . . . , j, the realizations ofλ T n on the path H . By setting Notice that the dispersion, computed as the sample standard deviation, is meant with respect to the mean value of theλ T n 's and not with respect to the "Poisson part" of the Hawkes intensity, λ 0 .
This measure allows us to get the general idea of how much the intensity values are far from λ 0 , but it does not explicitly quantify this distance. Specifically, M disp allows us to compare a Hawkes process with any Poisson process, regardless of its intensity, i.e. it answers the more general question whether a Poisson process exists fitting well with the observed phenomenon. In fact, the values ofλ T n may be concentrated around aλ much larger than λ 0 : the process cannot be approximated with a Poisson process with intensity λ 0 , but it may be close to a different Poisson process (the one with intensityλ). Otherwise, the values ofλ T n may be in average closer to λ 0 , but too sparse, so that neither the Poisson process with intensity λ 0 nor any Poisson process are a good model for the observed process. M disp is susceptible to large fluctuations of the values ofλ T n , i.e. M disp is large even if λ T n ≈ λ 0 for a high percentage of observations and few timesλ T n attains very large values.
In other words, M disp is more susceptible to the presence of large values ofλ T n −λ 0 than to the number of observations leading to a largeλ T n − λ 0 . For this reason, in some cases, it is discordant with the other measures; or rather we should say that, while other measures quantify an effect (e.g. the expected number of extra-Poisson jumps), which may even be quite weak, M disp catches the strength of a potential cause of the effect.

Integrated Intensity-Based Measure
Another measure of the departure of a process from a Poisson process can be derived by the integral of the "non-Poisson" term of intensity. In particular, in the case of a Hawkes process, the non-Poisson term coincides with the self-excitation term of the intensity. In order to compute this quantity, we need to fix a time horizon T and consider the process for t ∈ [0, T ].
(T ) − T λ 0 captures the mean excess of jumps of a Hawkes process with parameters (λ 0 , α, β) with respect to a Poisson process with intensity λ 0 . However, in order to make this quantity independent of the time horizon T and of the baseline intensity λ 0 , we divide it by T λ 0 obtaining Hence, along an observed path H , the expected number of non-Poisson jumps is M (H ) times the expected number of Poisson jumps.
(T ) is computed by applying its definition, as the integral of the conditional intensity λ t on [0, T ].
In view of its theoretical properties, we expect that M : -can detect the different behaviour of a Hawkes process from a Poisson process in those limit cases, when α is very small or β is very large, when commonly we conclude that the Hawkes process is very similar or tends to a Poisson, or quantify such a similarity; -is sensitive to clusters, that are a distinguishing feature of a Hawkes process.
We consider only the case of a non-explosive process, i.e. α < β. Theoretically, for a fixed α β , M is not affected by λ 0 . In fact, if we could assume stationarity, M = β β−α −1, regardless of the realized path of the process.
This fact may result counterintuitive, since one could expect that the larger the ratio α λ 0 (representing the jump impact on the intensity relatively to the baseline part λ 0 ), the larger the distance of the Hawkes process from the Poisson one. However, the closed formula for M in terms of α, β is not accurate when stationarity is not guaranteed, as it happens for processes on the half line. Also this measure, when computed on a Poisson process, would be equal to 0.

Measures Based on Inter-Arrival Times Frequency Distribution
It is known that inter-arrival times of a Poisson process are i.i.d. exponential r.v.'s. Interarrival times of a Hawkes process are not independent nor identically distributed (see Foschi et al. 2019, Proposition 2). We are interested however in studying how the values of the interarrival times are distributed with respect to the data generated by an exponential distribution with a given parameter. We would like to provide a synthetic and more precise quantification than the information given by a quantile-quantile plot, by measuring the intersection between the hypographs of the frequency distribution of the inter-arrival times and of the exponential density of parameter λ 0 . N (λ 0 ,α,β) ,

Definition 4 Given a realization H of the process
where f is the empirical frequency distribution of the inter-arrival times of H .

Remark 6
+∞ 0 min(f (x), λ 0 e −λ 0 x )dx is the area of the intersection between the hypographs of the two functions f (x) and λ 0 e −λ 0 x (see Fig. 1). Since the integral of a probability density function as well as the one of a relative frequency distribution is 1, Remark 7 When N is a Poisson process with intensity λ 0 , theoretically, for any realization H (that has an infinite number of jumps) M 0 ∩ (H ) = 0, and we fix this value as our benchmark. In practice, simulated paths or observed realizations of a process necessarily have a finite number of jumps, and the equality M 0 ∩ (H ) = 0 holds only as the limit when the number of jumps j tends to infinity. The convergence to 0 is due to the fact that, when j → +∞, the empirical density converges to the true theoretical density λ 0 e −λ 0 x .
In order to implement the computation of M 0 ∩ , an operational definition is needed. We denote by a, b respectively the minimum and the maximum observed inter-arrival time. It is not restrictive considering a = 0. We fix > 0, suitably small with respect to the length b − a, e.g. = 10 −4 .
where {x 1 , . . . , x j } are the inter-arrival times of the path H and e −λ 0 i ( We expect that, the stronger the self-excitation effect, the higher the number of interarrival times not following an exponential law, and therefore the smaller the intersection of the hypograph of their distribution with the one of an exponential distribution. Since M 0 ∩ compares the frequency distribution of the inter-arrival times with the exponential density with parameter λ 0 , it quantifies the distance of the Hawkes process from its "baseline" Poisson process.
This kind of measure is also suitable to be extended to the case when we don't know λ 0 or we aim at comparing the Hawkes process with a general Poisson process. In other words, M 0 ∩ can be adapted to measure how much the Poisson best fitting with data is actually well fitting. In this case, we estimate the parameterλ of the exponential best fitting with the inter-arrival times and, as above, we measure the intersection between the hypograph of the frequency distribution of the inter-arrival times and the one of the exponential density of parameterλ. We define M ∩ analogously to M 0 ∩ : Definition 5 Given a realization H of the process N (λ 0 ,α,β) , where f is the empirical frequency distribution of the inter-arrival times of H .

Remark 8
If H is a realization of a Poisson process with intensity λ 0 ,λ is very close to λ 0 and therefore M ∩ (H ) is approximatively equal to M 0 ∩ (H ). If H is a realization of a Poisson process with intensity λ = λ 0 andλ is the maximum likelihood estimator (MLE) of λ, the exponential density with parameterλ is the closest one to the empirical density of the inter-arrival times; thus M 0 ∩ (H ) ≥ M ∩ (H ). In particular, as j → +∞, M ∩ (H ) tends to 0, while M 0 ∩ (H ) does not. If, finally, H is a realization of a Hawkes process with parameters (λ 0 , α, β), since ∀ t > 0, λ t > λ 0 , the jumps of H are more frequent than the ones of a Poisson process with intensity λ 0 and thereforeλ > λ 0 . Even if the inter-arrival times have no more an exponential distribution, among the exponential densities, the one with parameterλ is the closest one to the empirical density of the inter-arrival times, and therefore M 0

Simulations Results and Comparisons
In this section, we illustrate by means of some examples the impact of different sets of parameters (λ 0 , α, β) on the four measures M disp , M , M 0 ∩ , M ∩ (H ) defined above. Intuitively a small α and a large β make the distances small; still we are interested in quantify it, in order to establish whether a Poisson process can be used to fit the data generated by N (λ 0 ,α,β) and possibly the error due to the approximation.
When α and β are both small or both large, we have no more general results to quantify the discrepancy of the Hawkes process from a Poisson one.
We provide examples of processes whose different behaviours may be explained by our measures. Since the difference from a Poisson process concerns several aspects, we expect that any trait has repercussions on a different measure and therefore that the four measures are not necessarily concordant.
For each process N (λ 0 ,α,β) with given parameters, we simulate d = 2500 paths H (λ 0 ,α,β) , on a time interval [0, T ], T =mδ. δ > 0 is a scale parameter, allowing us, in the applications, to express the time horizon T in the desired unit of measurement. We fix δ = 4.96 · 10 −5 . For different processes, instead, we choose differentm, in order to avoid realizations H (λ 0 ,α,β) with too few jumps. For each H (λ 0 ,α,β) , we compute the mean values of the four measures and their standard deviations, as reported in Table 1.

Description of
In a first group, we fix λ 0 = 3, β = 15 and vary α. As we can expect, as α increases, all the measures increase. In particular, the increase of M disp is a consequence both of wider intervals for the values ofλ T n 's and of the fact thatλ T n 's are less concentrated around the modal value, that is very close to λ 0 . This means that, for small α's the sequence {λ T n } n=1,...,j exhibits small and dense fluctuations, that become larger and less dense as α increases; furthermore for small α's, {λ T n } n=1,...,j comes back very often very close to λ 0 . This behaviour has also an impact on M : a {λ T n } n=1,...,j with small and dense fluctuations, often coming back to λ 0 has a hypograph with a smaller area, leading to a small M . Also the measures M 0 ∩ , M ∩ show that the processes are far from Poisson processes. We notice that, in the first two cases,λ is not significantly different from λ 0 and this implies that also M 0 ∩ and M ∩ are not significantly different. In the last two cases,λ > λ 0 significantly and this implies that M 0 ∩ > M ∩ ; but still M ∩ is significantly greater than 0. A wider case record can be observed in the second group of simulated processes, where we fix λ 0 = 20, α = 0.2 and vary β. As β increases, M disp , M and M 0 ∩ decrease. However, we notice that the decreasing trend of M 0 ∩ is significant only for small values of β.
For β = 0.3, {λ T n } n=1,...,j increases in a first period and then has fluctuations of a small amplitude with respect to the range of its values. Such range narrows as β increases, but, in the meanwhile, the relative amplitude of fluctuations and their frequency increases, until {λ T n } n=1,...,n often takes values very close to λ 0 (see also Fig. 2).
Finally, the last two groups in the table show how, even with a very small α (or α small with respect to λ 0 ), the measures are able to reveal that the processes are not Poisson. Table 2 Within each group, the values of M are not significantly different. This feature is consistent with the theoretical formula M = ( β α − 1) −1 , where M only depends on the ratio α β . The values also are close to the theoretical ones: M (N (λ 0 ,α,100α) ) = 0.0101, while M (N (λ 0 ,α,3α) ) = 0.5, that is in the confidence intervals of M (H (λ 0 ,α,3α) ) for the most part of the simulated processes, namely the ones with Table 1 The table summarizes the results obtained from the simulation of d = 2500 paths for each Hawkes process N (λ 0 ,α,β) with parameters λ 0 , α, β (reported in the first column) In the others columns are reported the mean of the measures M disp , M , M 0 ∩ , M ∩ and, before this last one, the mean of theλ. The standard errors of each measure are reported on the line below within brackets. m is the number of trials in each path for generating jumps α = 1, 5, 100. Again within the group with α β = 1 3 , for the ones with λ 0 = 300, we obtain not significantly different values of M 0 ∩ . The values of M ∩ are quite close each other, but display an increasing trend wrt α or β, while the values of M disp are strongly different and increasing wrt α or β. This means that, as α increases, the amplitude of the fluctuations of {λ T n } n=1,...,j increases too. However the larger amplitude is balanced by a faster decay, i.e. by a higher frequency of the fluctuations, making the area of the hypograph of {λ T n } n=1,...,j almost constant.

Description of
The use of the simulated paths has however a further application, to check whether, in the cases when a parameters' estimation is needed, the estimated values are correct.
In fact, apart from M ∩ , whose computation is completely based on data and does not involve the parameters, the measures depend on the values of λ 0 , α, β.
As an example, in Table 3 we show how the measures may be biased, when a wrong value is assigned to a parameter. Table 3 The corresponding values of the measures are significantly different from the values of the same measures computed on H (λ 0 ,α,β) (for some of them, see Table 1). This inconsistency informs us that (λ 0 ,α,β) is not a correct estimate of the parameters of the generating process.

Description of
We notice that, even if the wrong value of α,α, is very small (close to 0), the values of the measures are still significantly different from 0, meaning that they lead us to the correct conclusion that the process is not Poisson, when instead the parameters' estimate does not give us such an evidence. We also remark that M disp is not affected by a misspecification of  λ 0 , while M 0 ∩ is not affected by a misspecification of α. The value M disp deriving from the misspecifiedα is such that M disp M disp =α α . As to M 0 ∩ , it attains its minimum value, coinciding with M ∩ , forλ 0 =λ.

An Application to Data
We consider the sequence of jump times filtered out (see Foschi et al. 2019) from the dataset of the five minutes prices of the assets JPM, from 3/1/2006 to 31/7/2013. On a time horizon T = 7.53, corresponding tom = 151791, we obtain a record of 816 jump times, that can be described by an exponential Hawkes process. In Foschi et al. (2019) a procedure for parameters' estimation is also developed, providing us with the valuesλ 0 = 53.27,α = 4.72,β = 9.14. As mentioned, our first goal in computing M disp , M , M 0 ∩ , M ∩ is quantifying the discrepancy between a Poisson process and the sample H generated by the process N (53.27,4.72,9.14) with respect to different aspects. We obtain However, such measures allow us to achieve another important conclusion. We simulate, with the samem = 151791, d = 2500 paths H (53.27,4.72,9.14) of the Hawkes process N (53.27,4.72,9.14) and compute M disp (H (53.27,4.72,9.14) ) = 17.0054 (2.4148), M (H (53.27,4.72,9.14) ) = 1.0459 (0.0755), H (53.27,4.72,9.14) ) = 0.3813 (0.0098), λ = 111.5729 (3.1366), M ∩ (H (53.27,4.72,9.14) ) = 0.2853 (0.0159). We can now check whether the discrepancies obtained from the data are consistent with the reference values obtained from simulated paths. In this case, all the M(H )'s are not significantly different from the M(H (53.27,4.72,9.14) )'s and therefore we find a further confirmation of the fact that an exponential Hawkes model is well describing the data in H and that the parametersλ 0 = 53.27,α = 4.72,β = 9.14 are correctly estimated.

Concluding Remarks
After having theoretically studied the effect of jumps and their configurations on conditional or stochastic intensity and integrated intensity, we defined different measures, quantifying the distance between a Hawkes and a Poisson process. Since the difference between a Hawkes and a Poisson process is a complex matter and concerns several aspects, we need different measures, each one quantifying a different trait and having its own advantages and disadvantages. All our measures are designed to be applied to a dataset or to a single path generated by a stochastic process. The only exception is M , that, under some hypotheses, allows us to measure the theoretical discrepancy between a Hawkes data generating process N (λ 0 ,α,β) and a Poisson process with intensity λ 0 . On the other hand, M allows us to