Large Deviations for Hawkes Processes with Randomized Baseline Intensity

Abstract

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics.

1. Introduction

The Hawkes process is a self-exciting point process with a clustering effect and a jump rate that depends on its entire history [1]. There are many uses and benefits to Hawkes processes for modeling simple point processes. The Hawkes process exhibits self-exciting properties and clustering effects. Compared with a standard Poisson process, the intensity process for a point process consists of a summation of the baseline intensity and other terms that depend on the process’s past. Hawkes processes are typically used in the modeling of high-frequency trading to express temporal phenomena in a stochastic process that evolves in continuous time. In addition to capturing the self-exciting property and clustering effects, the Hawkes process is a natural generalization of the Poisson process. This process is a variable model that is amenable to statistical analysis. As a result, it has a wide range of applications in criminology [2], finance [3], seismology [4], DNA modeling [5], neuroscience [6], machine learning, and artificial intelligent [7,8]. A general description of the Hawkes process is given below.

Let N be a simple point process on $\mathbb{R}$ and let ${\mathcal{F}}_t^{-\infty }:=\sigma (N\left(C\right),C\in \mathcal{B}\left(\mathbb{R}\right),C\subset (-\infty ,t])$ be an increasing family of $\sigma -$ algebras. Any nonnegative ${\mathcal{F}}_t^{-\infty }$-progressively measurable process ${\lambda }_t$ having

$$\begin{array}{c}\hfill E\left[N(a,b]|{\mathcal{F}}_a^{-\infty }\right]=E\left[{\int }_a^b{\lambda }_{s}ds|{\mathcal{F}}_a^{-\infty }\right]\end{array}$$

(1)

$\mathrm{a}.\mathrm{s}.$ for all intervals $(a,b]$ is called an ${\mathcal{F}}_t^{-\infty }$-intensity of N. We use the notation $N_t:=N(0,t]$ to denote the number of points in the interval $(0,t].$

Let N be a simple point process containing an ${\mathcal{F}}_t^{-\infty }$-intensity (general Hawkes process)

$$\begin{array}{c}\hfill {\lambda }_t:=\lambda \left({\int }_{-\infty }^{t}h(t-s)N\left(ds\right)\right),\end{array}$$

(2)

where $\lambda (·):{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is locally integrable and left continuous, $h(·):{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and assuming ${\parallel h\parallel }_{L^1}={\int }_0^{\infty }h\left(t\right)dt<\infty .$ Here, ${\int }_{-\infty }^{t}h(t-s)N\left(ds\right)$ stands for ${\int }_{(-\infty ,t)}h(t-s)N\left(ds\right).$ It is assumed that $N(-\infty ,0]=0,$ in other words, the history of the Hawkes process is nil. Typically, $h(·)$ and $\lambda (·)$ are known as the exciting function and rate function, respectively. The Hawkes process can either be linear or nonlinear. It is linear only when $\lambda (·)$ is linear, otherwise, it is nonlinear. The above-mentioned model is non-Markovian. This is because the future evolution of this model depends upon its history of events and their occurrences. It is only in a special case that the above model behaves as Markovian. The Hawkes process has several applications, such as in criminology [2], finance [3], seismology [4], DNA modeling [5], and neuroscience [6]. Since it has both self-exciting and clustering properties, it is considered appropriate for certain financial applications. The clustering and self-exciting properties of the Hawkes processes have been utilized in modeling correlated defaults and estimating credit derivatives in finance; see Errais et al. [3] and Dassios and Zhao [9].

The linear Hawkes process [1] is much more commonly used than the non-linear one. Almost all applications of the Hawkes process use the linear model, such as immigration-birth representation [10], central limit theorems, the law of large numbers, large deviation principles, and the Bartlett spectrum [1,11,12,13]. In contrast, the nonlinear Hawkes process finds less application due to the lack of immigration birth representation and computational tractability. The nonlinear model was first reported by Brémaud and Massoulié [14] and has since been studied extensively [15,16,17,18,19], for example, the central limit theorem in [17], the large deviation principles [16], and applications in financial mathematics [19,20]. Jaisson and Rosenbaum presented limit behavior theorems and scaling limits for the rough fractional diffusion of the nearly unstable Hawkes model [21,22]. Seol studied the inverse process of Hawkes processes and reported limit theorems for the arrival time ${\tau }_n$, arrival time [23]. Due to the continuous progress in storage technology, data-driven models have gained significant attention recently. However, the Hawkes process is defined in continuous-time settings, whereas events, in reality, are often recorded in discrete time. More importantly, the data should be collected in fixed intervals to avoid showing aggregate results. For instance, continuous-time Hawkes processes can model events occurring at unequal time intervals, whereas equal interval discrete-time Hawkes processes are required to model events occurring at fixed intervals. Seol proposed a 0–1 discrete case Hawkes process starting from an empty history and used it to study central limit theorems, laws of large numbers, and invariance principles [24]. Recently, Wang [25,26] studied the limit behaviors of a discrete-time Hawkes process with random marks and proved the large and moderate deviations for a discrete-time Hawkes process with marks.

Seol also investigated moderate deviation principles for Hawkes processes with random marks [27], and limit laws for the process of the compensator in Hawkes processes [28]. In addition to large time limits, Gao and Zhu also conducted studies on asymptotic results [29,30,31,32]. Furthermore, several studies related to the extension and modification of the classical Hawkes process have been reported in the literature. For example, in one study, the baseline intensity was considered to be inhomogeneous in time [33]. In another report, a dynamic contagion model was used where the immigrants were assumed to arrive following the Cox process with shot noise intensity [9]. Next, Wheatley et al. considered the renewal Hawkes process, which is a generalization of the classical Hawkes process, where immigrants arrive according to a renewal process instead of a Poisson process [34]. In addition to the studies mentioned above, other variations and extensions of the Hawkes process have also been reported in the literature [9,35,36,37,38]. Recently, Seol introduced the inverse Markovian Hawkes process and studied the limit theorems for the same. The inverse Markovian Hawkes process has the features of several existing models of the self-exciting process [39]. Seol also reported an extended version of the inverse Markovian Hawkes model [40] and non-Markovian inverse model [41]. In the present paper, linear Hawkes processes are employed with randomized baseline intensity, which has important applications. We also studied the limit theorems for the randomized baseline intensity of the Hawkes process. It is worth mentioning that Gao and Zhu studied the functional central limit theorems for the stationary case of the Hawkes process with its application to infinite-server queues [33]. They also studied large baseline intensity asymptotics. However, there are two main differences between their paper [33] and the current paper. First, in the present work, empty history was used, but Gao and Zhu used the stationary version of the Hawkes process [33]. Second, in the current paper, we studied scalar limit theorems for fixed t, whereas Gao and Zhu [33] considered functional limit theorems. The present paper is organized as follows: Section 2 presents some auxiliary results to support our main results, Section 3 presents our main results, and Section 4 contains the proofs of the main theorems.

2. Preliminaries

A review of the main problems is given in this section, along with an introduction to the classical results. Throughout the paper, we will be using a set of assumptions.

Assumption A1.

(1) $\lambda \left(z\right)=\nu +z$, for some $\nu >0,$

(2) ${\parallel h\parallel }_{L^1}<1$ where ${\parallel h\parallel }_{L^1}={\int }_0^{\infty }h\left(t\right)dt<\infty .$

Assumption A1 denotes that $\lambda$ is a linear and increasing function. This implies that the Hawkes process represents immigration-birth very well, which is the main tool in the current paper.

A linear Hawkes process can be represented by an immigration-birth process, which is used in proving the theorem. The following is a general description of the immigration-birth representation. An immigrant arrives following a standard homogeneous Poisson process at a constant intensity $\nu >0$. Then, based on the Galton–Watson tree, each immigrant generates a number of children [10]. Let $\eta$ be the number of an immigrant’s children, and $\eta$ has a Poisson distribution with parameter ${\parallel h\parallel }_{L^1}$. Conditioned on the number of children of an immigrant, the time when a child is born has a probability density function of

$$\begin{array}{c}\hfill \frac{h(·)}{{\parallel h\parallel }_{L^1}}.\end{array}$$

(3)

The immigration-birth representation signifies that $N_t$ denotes the total number of immigrants and their descendants up to time t.

Some reviews on the results of the Hawkes processes are shown below.

Limit Theorems for Hawkes Processes

The limit theorems of both linear and nonlinear Hawkes processes have been studied by many researchers.

Linear case model: Since $\lambda (·)$ is linear, say $\lambda \left(z\right)=\nu +z$ for some $\nu >0,$ and ${\parallel h\parallel }_{L^1}<1,$ there is an excellent immigration-birth representation, and the limit theorems can be represented more explicitly. Daley and Vere-Jones [42] investigated the proof of the law of large numbers for the linear case of the Hawkes process. Functional central limit theorems for linear cases of multivariate Hawkes processes with certain assumptions were studied by Bacry et al. [11]. Bordenave and Torrisi [12] proved that if ${0<\parallel h\parallel }_{L^1}<1$ and ${\int }_0^{\infty }th\left(t\right)dt<\infty ,$ then $(\frac{N_t}{t}\in ·)$ satisfies the large deviation principles. Zhu reported the moderate deviation principle for the linear case in continuous-time Hawkes processes [18], as well as the limit theorems for linear Hawkes processes with random marks [37].

Nonlinear case model: Since $\lambda (·)$ is nonlinear, the usual immigration-birth representations are no longer relevant. This makes the nonlinear model much harder to study. Brémaud and Massoulié [14] proved that there are unique stationary versions of nonlinear Hawkes processes, under certain conditions, and nonstationary versions converge to equilibrium. The central limit theorem obtained by Zhu [15,17] demonstrated the large deviation for a special case of the nonlinear model when $h(·)$ is exponential or a sum of exponentials. Zhu [16] proved a process-level large deviation principle for nonlinear Hawkes processes with general $h(·)$, namely the level-3 large deviation principle and, therefore, by the contradiction principle, the level-1 large deviation principle for $(\frac{N_t}{t}\in ·)$. Before we proceed, let us review some limit theorem results of the linear Hawkes process from the literature. Daley and Vere-Jones [42] proved the law of large numbers for the linear Hawkes process, as follows.

$$\begin{array}{c}\hfill \frac{N_t}{t}\to \frac{\nu }{1-{\parallel h\parallel }_{L^1}} \mathrm{as} t\to \infty \end{array}$$

(4)

The functional central limit theorem for the linear multivariate case of the Hawkes process, under certain assumptions, was obtained by Bacry et al. [11]. They proved that

$$\begin{array}{c}\hfill \frac{N_{·t}-·\mu t}{\sqrt{t}}\to \sigma B(·), \mathrm{as} t\to \infty ,\end{array}$$

(5)

where $B(·)$ is a standard Brownian motion and

$$\begin{array}{c}\hfill \mu =\frac{\nu }{1-{\parallel h\parallel }_{L^1}} \mathrm{and} {\sigma }^2=\frac{\nu }{{(1-\parallel h\parallel }_{L^1}{)}^3}\end{array}$$

(6)

The convergence used in the above theorem is weak convergence on the space $D[0,1],$ which consists of càdlàg functions on $[0,1],$ and is equipped with the Skorokhod topology. Bordenave and Torrisi [12] proved that if ${0<\parallel h\parallel }_{L^1}<1$ and ${\int }_0^{\infty }th\left(t\right)dt<\infty ,$ then $(\frac{N_t}{t}\in ·)$ satisfies the large deviation principle with the good rate function $I(·),$ which means that for any closed set $C\subset \mathbb{R},$

$$\underset{t\to \infty }{lim\; sup}\frac{1}{t}log\mathbb{P}(N_t/t\in C)\le -\underset{x\in C}{inf}I\left(x\right),$$

and for any open set $G\subset \mathbb{R},$

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; inf}\frac{1}{t}log\mathbb{P}(N_t/t\in G)\ge -\underset{x\in G}{inf}I\left(x\right),\end{array}$$

(7)

where

$$\begin{array}{c}\hfill I\left(x\right)=\left\{\begin{array}{cc}x{\theta }_x+\nu -\frac{\nu x}{\nu +{\parallel h\parallel }_{L^1}x}\hfill & \mathrm{if} x\in (0,\infty )\hfill \\ \nu \hfill & \mathrm{if} x=0\hfill \\ +\infty \hfill & \mathrm{if} x\in (-\infty ,0).\hfill \end{array}\right.\end{array}$$

(8)

where $\theta ={\theta }_x$ is the unique solution in $(-\infty ,\parallel h{\parallel }_{L^1}-1-log\parallel h{\parallel }_{L^1}),$ of

$$\begin{array}{c}\hfill \mathbb{E}\left(e^{\theta S}\right)=\frac{x}{\nu +{x\parallel h\parallel }_{L^1}},x>0\end{array}$$

(9)

where S in the above equation denotes the total number of descendants of an immigrant, including the immigrant himself.

The rate function described above $I\left(x\right)$ can be represented as a more explicit form. Note that (see [43] for detail), for all $\theta \in (-\infty ,\parallel h{\parallel }_{L^1}-1-log\parallel h{\parallel }_{L^1}),$$\mathbb{E}\left(e^{\theta S}\right)$ satisfies

$$\begin{array}{c}\hfill \mathbb{E}\left(e^{\theta S}\right)=e^{\theta }{e}^{{\parallel h\parallel }_{L^1}(\mathbb{E}\left(e^{\theta S}\right)-1)},\end{array}$$

(10)

which implies that ${\theta }_x=log\left(\frac{x}{\nu +{x\parallel h\parallel }_{L^1}}\right)-{\parallel h\parallel }_{L^1}\left(\frac{x}{\nu +{x\parallel h\parallel }_{L^1}}-1\right).$ Substituting into the formula, we have

$$\begin{array}{c}\hfill I\left(x\right)=\left\{\begin{array}{cc}xlog\left(\frac{x}{\nu +{x\parallel h\parallel }_{L^1}}\right)-x+{\parallel h\parallel }_{L^1}x+\nu \hfill & \mathrm{if} x\in (0,\infty )\hfill \\ \nu \hfill & \mathrm{if} x=0\hfill \\ +\infty \hfill & \mathrm{if} x\in (-\infty ,0).\hfill \end{array}\right.\end{array}$$

(11)

Zhu [18] proved that if ${\parallel h\parallel }_{L^1}<1$ and ${sup}_{t>0}{t}^{3/2}h\left(t\right)\le C<\infty ,$ then for any Borel set A and time sequence $\sqrt{n}\ll c\left(n\right)\ll n,$ there exists a moderate deviation principle

$$\begin{array}{c}-\underset{x\in A^{\circ }}{inf}J\left(x\right)\le \underset{t\to \infty }{lim\; inf}\frac{t}{c{\left(t\right)}^2}log\mathbb{P}\left(\frac{1}{c\left(t\right)}(N_t-\mu t)\in A\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \underset{t\to \infty }{lim\; sup}\frac{t}{c{\left(t\right)}^2}log\mathbb{P}\left(\frac{1}{c\left(t\right)}(N_t-\mu t)\in A\right)\le -\underset{x\in \overline{A}}{inf}J\left(x\right)\end{array}$$

(12)

where $J\left(x\right)=\frac{x^2{(1-\parallel h\parallel }_{L^1}{)}^3}{2\nu }.$

3. Statement of the Main Results

This section states the main results of this paper. First, the asymptotic results for the randomized baseline intensity will be presented. The results of the large deviation principle of the Hawkes process with the randomized baseline intensity are shown below.

A Hawkes process $N_t$ has the following intensity:

$${\lambda }_t=\nu +{\int }_0^{t-}h(t-s)d{N}_s,$$

(13)

where $h(·)$ is the exciting function, $\nu >0$ is the baseline intensity.

We consider a Hawkes process $N_t$ with randomized baseline intensity, i.e., $\nu$ follows a distribution on $[{\nu }_{min},{\nu }_{max}]$, with $0<{\nu }_{min}<{\nu }_{max}<\infty$ and a probability density function $p\left(\nu \right)>0$ for every $\nu \in [{\nu }_{min},{\nu }_{max}]$. Conditional on $\nu$, $N_t$ has the process of intensity

$${\lambda }_t=\nu +{\int }_0^{t-}h(t-s)d{N}_s.$$

(14)

For constant $\nu$, we know that $\mathbb{P}(N_t/t\in ·)$ satisfies the large deviation principle with the function of the rate

$$I(x;\nu )=\left\{\begin{array}{cc}xlog\left(\frac{x}{\nu +{x\parallel h\parallel }_{L^1}}\right)-x+x{\parallel h\parallel }_{L^1}+\nu \hfill & \mathrm{if}x\in [0,\infty )\hfill \\ +\infty \hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

(15)

In this paper, we consider a modified model of the Hawkes process with randomized baseline intensity, and investigate whether $\mathbb{P}(N_t/t\in ·)$ satisfies a large deviation principle with a rate function that depends on the randomized baseline intensity, as shown below.

Theorem 1. 

Considering Assumption 1 is satisfied, $\mathbb{P}(N_t/t\in ·)$ satisfies the large deviation principle with rate function:

$$I\left(x\right)=\left\{\begin{array}{cc}I(x;{\nu }_{max})\hfill & ifx>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}},\hfill \\ 0\hfill & ifx\in [\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}],\hfill \\ I(x;{\nu }_{min})\hfill & if0\le x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\hfill \end{array}\right.$$

(16)

and $I\left(x\right)=+\infty$ otherwise.

4. The Proofs of the Theorems

This section furnishes the proofs of the main theorems. The following lemma plays a key role in proving the main theorem.

Lemma 1. 

For any $x>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=-I(x;{\nu }_{max}).$$

(17)

Proof. 

For every $x>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}$,

$$\begin{array}{cc}\hfill \mathbb{P}(N_t\ge xt)& \ge \mathbb{P}(\nu \ge {\nu }_{max}-ϵ)\mathbb{P}(N_t\ge xt|\nu \ge {\nu }_{max}-ϵ)\hfill \\ \hfill & \ge {\int }_{{\nu }_{max}-ϵ}^{{\nu }_{max}}p\left(\nu \right)d\nu \mathbb{P}(N_t\ge xt|\nu ={\nu }_{max}-ϵ).\hfill \end{array}$$

Thus,

$$\underset{t\to \infty }{lim\; inf}\frac{1}{t}log\mathbb{P}(N_t\ge xt)\ge -I(x;{\nu }_{max}-ϵ).$$

(18)

On the other hand,

$$\mathbb{P}(N_t\ge tx)\le \mathbb{P}(N_t\ge tx|\nu ={\nu }_{max}),$$

(19)

and, thus,

$$\underset{t\to \infty }{lim\; sup}\frac{1}{t}log\mathbb{P}(N_t\ge xt)\le -I(x;{\nu }_{max}).$$

(20)

Since (20) holds for any sufficiently small $ϵ>0$, we conclude that for $x>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}(N_t\ge xt)=-I(x;{\nu }_{max}).$$

(21)

This implies that for any $x>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=-I(x;{\nu }_{max}).$$

(22)

This completes the proof of Theorem 1. □

Lemma 2. 

For any $x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=-I(x;{\nu }_{min}).$$

(23)

Proof. 

For every $x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}}$,

$$\begin{array}{cc}\hfill \mathbb{P}(N_t\ge xt)& \le \mathbb{P}(\nu \ge {\nu }_{min}-ϵ)\mathbb{P}(N_t\ge xt|\nu \ge {\nu }_{min}-ϵ)\hfill \\ \hfill & \le {\int }_{{\nu }_{min}-ϵ}^{{\nu }_{min}}p\left(\nu \right)d\nu \mathbb{P}(N_t\ge xt|\nu ={\nu }_{min}-ϵ).\hfill \end{array}$$

Thus,

$$\underset{t\to \infty }{lim\; inf}\frac{1}{t}log\mathbb{P}(N_t\ge xt)\le -I(x;{\nu }_{min}-ϵ).$$

(24)

On the other hand,

$$\mathbb{P}(N_t\ge tx)\ge \mathbb{P}(N_t\ge tx|\nu ={\nu }_{min}),$$

(25)

and, thus,

$$\underset{t\to \infty }{lim\; sup}\frac{1}{t}log\mathbb{P}(N_t\ge xt)\ge -I(x;{\nu }_{min}).$$

(26)

Since (26) holds for any sufficiently small $ϵ>0$, we conclude that for $x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}(N_t\ge xt)=-I(x;{\nu }_{min}).$$

(27)

This implies that for any $x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=-I(x;{\nu }_{min}).$$

(28)

This completes the proof of Theorem 2.

Similarly, we can show that for any $x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}}$,

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=-I(x;{\nu }_{min}).$$

(29)

This completes the proof of Theorem 2. □

Lemma 3. 

For any $x\in [\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}]$, we have

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=0.$$

(30)

Proof. 

Finally, for any $x\in [\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}]$, we have for any sufficiently small $\delta >0$,

$$\begin{array}{cc}\hfill \mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)& \ge \mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)|\nu \in B_{\delta (1-\parallel h{\parallel }_{L^1})}\left(x\right(1-{\parallel h\parallel }_{L^1}\left)\right)\cap [{\nu }_{min},{\nu }_{max}]\right)\hfill \\ \hfill & ·\mathbb{P}\left(\nu \in B_{\delta (1-\parallel h{\parallel }_{L^1})}\left(x\right(1-{\parallel h\parallel }_{L^1}\left)\right)\cap [{\nu }_{min},{\nu }_{max}]\right).\hfill \end{array}$$

Conditional on ${(x-\delta )(1-\parallel h\parallel }_{L^1}{)\le \nu \le (x+\delta )(1-\parallel h\parallel }_{L^1})$,

$$\underset{t\to \infty }{lim\; sup}\frac{N_t}{t}\le x+\delta ,$$

(31)

${\mathbb{P}(·|(x-\delta )(1-\parallel h\parallel }_{L^1}{)\le \nu \le (x+\delta )(1-\parallel h\parallel }_{L^1}\left)\right)$-a.s. and

$$\underset{t\to \infty }{lim\; inf}\frac{N_t}{t}\ge x-\delta ,$$

(32)

${\mathbb{P}(·|(x-\delta )(1-\parallel h\parallel }_{L^1}{)\le \nu \le (x+\delta )(1-\parallel h\parallel }_{L^1}\left)\right)$-a.s.

This implies that

$$\underset{\delta \to 0}{lim}\underset{t\to \infty }{lim}\frac{1}{t}log\mathbb{P}\left(\frac{N_t}{t}\in B_{\delta }\left(x\right)\right)=0.$$

(33)

This completes the proof of Theorem 3. □

Proof of Theorem 1. 

Finally, for any $K>0$,

$$\underset{t\to \infty }{lim\; sup}\frac{1}{t}log\mathbb{P}(N_t\ge K)\le -I(K;{\nu }_{max}),$$

(34)

which goes to $-\infty$ as $K\to \infty$.

Hence, with Lemmas 1–3 and the fact that (34) is satisfied, we conclude that $\mathbb{P}(N_t/t\in ·)$ satisfies a large deviation principle with the rate function:

$$I\left(x\right)=\left\{\begin{array}{cc}I(x;{\nu }_{max})\hfill & \mathrm{if}x>\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}},\hfill \\ 0\hfill & \mathrm{if}x\in [\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\frac{{\nu }_{max}}{1-{\parallel h\parallel }_{L^1}}],\hfill \\ I(x;{\nu }_{min})\hfill & \mathrm{if}0\le x<\frac{{\nu }_{min}}{1-{\parallel h\parallel }_{L^1}},\hfill \end{array}\right.$$

(35)

and $I\left(x\right)=+\infty$ otherwise. This completes the proof of Theorem 1. □

5. Discussion

Hawkes processes are generally defined for continuous-time settings as self-exciting simple point processes with a clustering effect, and their jump rate depends on the entire history of the process. Due to the fact that past events determine the future developments of self-exciting point processes, the Hawkes model is generally non-Markovian. However, under special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. Difficulties arise in the case of non-Markovian processes when the exciting function is not an exponential function or a sum of exponential functions. Many disciplines use point processes to analyze data, including criminology [2], finance [3], seismology [4], DNA modeling [5], neuroscience [6], and many others. There are a number of point processes, including the Poisson method, whose increments have independent timing, but real-life data generally do not demonstrate independent timing. Despite its importance for applications, many key theoretical results in the area remain unknown, including self-excitation, clustering, and contagion. In addition to capturing both the self-exciting property and clustering effect, the Hawkes process is a natural generalization of the Poisson process. Statistical analysis is possible for this process since it is variable. The large deviation theory describes the probability of rare events that depend on a parameter, which is one of the important techniques used to deal with some probabilistic properties. Moreover, the large deviation result is helpful in studying the ruin probabilities of a risk process when the claim arrivals follow a Hawkes process. The intensity process is one of the main methods used for studying the dynamic properties of general point processes, and is particularly important for credit risk analysis. The intensity of this point process is given by the sum of a baseline intensity plus other terms that depend on the entire history of the point process, compared with a standard Poisson process. The baseline intensity is usually defined for deterministic cases and plays a crucial role in the Hawkes process. In this paper, we consider a linear Hawkes model when the baseline intensity is randomly defined. Compared with previous results in the literature [19], we investigate several asymptotic results to observe financial applications. In particular, we have proven asymptotic results of the large deviation principle for the newly defined model, which has wide applications in insurance, finance, queue theory, and statistics. Therefore, this study has contributed to a better understanding of the large-time behavior of self-exciting point processes and provided numerous theoretical asymptotic behaviors of the Hawkes model, such as the large deviation principles for the randomized baseline intensity model, which was newly introduced in this article. In the future, we plan to perform additional experiments to verify the reliability of the Hawkes process model proposed in this article and consider more complex models that can be newly defined by the intensity processes of Hawkes models.

6. Conclusions

The Hawkes process, introduced by Hawkes [1], is a self-exciting point process, which means that future arrivals are increased by each arrival for a certain period of time. This process is an important class of the stochastic process, which found wide application in diverse areas, such as criminology [2], finance [3], seismology [4], DNA modeling [5], and neuroscience [6]. Recently, the Hawkes process was applied in machine learning and artificial intelligence [7,8]. In general, the model described in the current paper is non-Markovian. In the non-Markovian model, a self-exciting system with future evolution is controlled by the timing of past events. On the other hand, the Markovian model is used in special cases. In other words, the Hawkes process depends on the entire history and has a long memory. In addition to its self-exciting properties, it has clustering properties, which makes it appealing to applications in the financial industry. Although several limit theorems for linear and nonlinear Markovian cases are well known, there has been relatively little study of non-Markovian and inverse Hawkes processes. In a recent paper, Seol [39] presented a new inverse Markovian Hawkes process, mentioning the differences between the general Hawkes process and the inverse Markovian Hawkes process. The Hawkes process assumes that if there have been many jumps in the past, there will be more jumps in the future as well. Inverse Hawkes processes, however, predict larger jumps in the future as more jumps occurred in the past. Hawkes processes exhibit self-excitation that is dependent on the intensity of the process, whereas inverse Hawkes processes exhibit self-excitation that is related to the size of the jumps. Therefore, while the Hawkes process represents self-excitation in terms of frequency, the inverse Hawkes process represents self-excitation in terms of severity. The extended version of the inverse Markovian Hawkes process was introduced by Seol [40] and he studied several theoretical results. One of the best tools used for studying the dynamical properties of a general point process is the intensity process. In particular, the intensity process is important for studying credit risk. In the current paper, a more general (but active) model of the Hawkes process with randomized baseline intensity was considered. The large deviations principle for Hawkes processes with randomized baseline intensity was obtained in Theorem 1. Our results are important for various financial applications and add to the existing literature. Additionally, the findings of this paper have a broader impact on society beyond finance. In this paper, researchers and the general public can gain a deeper understanding of how complex systems cluster and self-excite. By using these techniques, we will build on, enhance, and better understand topical applications of self-exciting point processes. The general public and academics will be motivated to use them in the future. The newly defined model in this paper can be extended to other asymptotic results, such as the law of large numbers, central limit theorems, and moderate deviations in the future. In addition, we will consider the renewal Hawkes process and dynamic contagion model with several asymptotic theorems as potential projects.