2.1. The continuous-time setting

For a fixed $T>0$ , let h be a function in $L^1([0,T])$ . Let $\psi$ be a positive L-Lipschitz function on $\mathbb{R}$ and b be a positive Borel function on $\mathbb{R}$ . We now state the stability assumption on the aforementioned elements.

Assumption 1. We have

\begin{align*}\rho_h\;:\!=\;L\|h\|_1\mathbb{E}[b(Y)]<1,\end{align*}

where Y is a random variable of distribution $\nu$ and $\|h\|_1= \int_0^T |h(t) |\textrm{d} t.$ We also assume that $\nu$ has a finite first moment; that is, $\mathbb{E} [|Y|] <+\infty$ .

We now give the definition of the marked compound Hawkes process (or risk) as a result of thinning from the underlying tridimensional Poisson measure P. The procedure is now standard (see, for example, [Reference Brémaud and Massoulié6] or [Reference Ogata33]) and gives the Hawkes process as the unique pathwise solution of a stochastic differential equation (SDE) as examined in more detail in [Reference Hillairet, Réveillac and Rosenbaum21].

Definition 1. Let P be a tridimensional Poisson measure on $\mathbb{R}_+^2 \times \mathbb{R}$ of intensity $\textrm{d} t \times \textrm{d}\theta \times \nu(\textrm{d} y)$ . Fix the time horizon $T>0$ and let $h \in L^1 ([0,T])$ , $\psi\;:\; \mathbb{R} \to \mathbb{R}_+$ $L$ -Lipschitz, and $b\;:\;\mathbb{R} \to \mathbb{R}_+$ such that Assumption 1 is in force. The SDE on [0,T]

(3) \begin{equation} \begin{cases} R_t &=\int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} y\mathbf 1_{\theta \leq \lambda _s} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y),\\[5pt] \lambda_t &=\psi \left( \int_{[0,t) \times \mathbb{R}_+ \times \mathbb{R}} h(t-s) \mathbf 1_{\theta \leq \lambda _s} b(y)P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y) \right) \end{cases}\end{equation}

has a unique pathwise solution $(R, \lambda)$ such that R is $\mathcal F$ measurable and $\lambda \in L^1(\textrm{d}\mathbb P \times \textrm{d} t)$ is $\mathcal F$ predictable.

We say that R is a marked Hawkes risk of kernel h, jump rate $\psi$ , and claim size distribution $\nu$ . We call $\lambda$ the intensity of R, and b the marks the modulation function.

Furthermore, we define the marked simple Hawkes process N as

\begin{align*}N_t \;:\!=\;\int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} \mathbf 1_{\theta \leq \lambda _s} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y),\;\;t\in [0,T],\end{align*}

and the auxiliary process as

\begin{align*}\xi_t\;:\!=\;\int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} b(y)\mathbf 1_{\theta \leq \lambda _s} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y).\end{align*}

Proof. The proof is an adaptation of a classical contraction argument to the $L^1$ setting, following the proof of Theorem 1.2 in [Reference Graham18].

First, we start by mentioning that the system (3) is triangular. Therefore, once $\lambda$ is constructed, R is simply obtained by thinning from the underlying Poisson measure P. We thus focus on the construction of the intensity $\lambda$ .

Let f be the function that maps a predictable, integrable process on [0, T] of finite expectation to the integrable predictable process $\kappa\;:\!=\;f(\lambda)$ defined by

\begin{equation*} \kappa_t=\psi \left( \int_{[0,t) \times \mathbb{R}_+ \times \mathbb{R}} h(t-s) \mathbf 1_{\theta \leq \lambda _s} b(y)P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y) \right)\!. \end{equation*}

Given two predictable processes $\lambda$ and $\lambda'$ and their images $\kappa$ and $\kappa'$ , we have

\begin{align*} \mathbb{E} \left [\int_{0}^T \left |\kappa'_t-\kappa_t\right| \textrm{d} t\right]& \leq L \mathbb{E} \left[ \int_0^T \left |\int_{[0,t)\times \mathbb{R}_+\times \mathbb{R} } h(t-s)(\mathbf 1 _{\theta \leq \lambda'_s}-\mathbf 1_{\theta \leq \lambda_s})b(y)P(\textrm{d} s, \textrm{d} \theta , \textrm{d} y) \right | \textrm{d} t\right]\\[5pt] &\leq L \mathbb{E} \left[ \int_0^T\int_{[0,t)\times \mathbb{R}_+\times \mathbb{R} } |h(t-s)| \left |\mathbf 1 _{\theta \leq \lambda'_s}-\mathbf 1_{\theta \leq \lambda_s}\right |b(y)P(\textrm{d} s, \textrm{d} \theta , \textrm{d} y) \textrm{d} t\right ]\!, \end{align*}

and using the fact that $\lambda$ and $\lambda'$ are predictable, we have

\begin{align*} \mathbb{E} \left [\int_{0}^T \left |\kappa'_t-\kappa_t\right| \textrm{d} t\right]& \leq L \mathbb{E} \left[ \int_0^T\int_{[0,t)}|h(t-s)| \left | \lambda'_s-\lambda_s\right | \mathbb{E} [b(Y)]\textrm{d} s \textrm{d} t\right]\\[5pt] &= L\mathbb{E} [b(Y)]\mathbb{E} \left[ \int_0^T\int_s^T |h(t-s)| \left | \lambda'_s-\lambda_s\right | \textrm{d} t \textrm{d} s\right]\\[5pt] &=L\mathbb{E} [b(Y)]\mathbb{E} \left[ \int_0^T \left | \lambda'_s-\lambda_s\right |\int_s^T |h(t-s)| \textrm{d} t \textrm{d} s\right]\\[5pt] &\leq L\mathbb{E} [b(Y)] \|h\|_1 \mathbb{E} \left[\int_0^T \left|\lambda'_s-\lambda_s \right | \textrm{d} t \right]\!. \end{align*}

By introducing the norm

\begin{align*}\|X\|=\mathbb{E} \left[ \int_0^T|X_t|\textrm{d} t\right ]\!,\end{align*}

we can write the last inequality in the form

\begin{align*}\|f(\lambda')-f(\lambda)\| \leq L \mathbb{E} [b(Y)] \|h\|_1 \|\lambda '-\lambda\|,\end{align*}

where $L \mathbb{E} [b(Y)] \|h\|_1 <1$ thanks to Assumption 1. Using the Banach fixed-point theorem, we conclude that f has a unique fixed point $\lambda$ .

If $\psi(x)=\mu+ x, x \in \mathbb{R}_+$ for a positive constant $\mu$ under the constraint $h\geq 0$ , we say that the Hawkes process/risk is linear. In this particular case, the Hawkes dynamics have a branching process representation. Although the first two moments of the process are explicitly known (up to the computation of an infinite sum of convolutions of h; see [Reference Bacry, Delattre, Hoffmann and Muzy3, Reference Hillairet, Réveillac and Rosenbaum21]), linear Hawkes processes do not allow for self-inhibition.

The marked Hawkes risk can also be defined in a more elementary (yet informal) way without explicit use of the thinning procedure. Let N be a simple point process on [0, T] and $(Y_k)_{k\in \mathbb N}$ be a family of independent and identically distributed (i.i.d) random variables of common distribution $\nu$ . Note that we do not assume the variables $(Y_k)_{k\in \mathbb N}$ and the process N to be independent. N is said to be a marked simple Hawkes process if its intensity $\lambda$ follows the dynamics

\begin{align*}\lambda_t = \psi \left( \sum_{\tau_i<t} h(t-\tau_i) b(Y_i)\right )\!,\end{align*}

where $(\tau_i)_{i\in \mathbb N}$ are the arrival times of the points of N. Note that the intensity can also be expressed in the integral form

\begin{align*}\lambda_t = \psi \left( \int_{0}^{t-} h(t-s) \textrm{d} \xi_s\right )\!,\end{align*}

where $\xi$ is the auxiliary process $\xi_t=\sum_{k=1}^{N_t}b(Y_k),\;\;t\in [0,T]$ . The risk process is simply the aggregation of all the marks

\begin{align*}R_t=\sum_{k=1}^{N_t} Y_k,\;\;t\in [0,T].\end{align*}

The marked Hawkes process is useful in modeling phenomena where the intensity is impacted not only by the realization of an event $\tau_i$ but also by its ‘severity’ $Y_i$ . For instance, the choice $b(y)= \mathbf 1 _{y\geq a}$ means that only claims of a size larger than a given threshold a have an impact on the intensity. Karabash and Zhu [Reference Karabash and Zhu24] provided limit theorems for a general class of marked Hawkes processes, albeit in the linear setting.

This constitutes a generalization of the compound Hawkes process usually studied in the literature (see [Reference Errais, Giesecke and Goldberg13, Reference Khabou, Privault and Réveillac26, Reference Khabou and Torrisi28]) where the marks Y do not impact the intensity.

If the modulation function b is chosen to be equal to the constant 1, we can retrieve the usual unmarked intensity $\lambda_t=\psi\left(\int_0^{t-}h(t-s) \textrm{d} N_s \right)$ . Similarly, the choice $Y \equiv 1$ ensures that $R\equiv N$ ; hence we will focus exclusively on R.

The goal is to suggest an intuitive discretization scheme on [0, T] and to obtain a bound on the distance between this scheme and the continuous-time process in a convenient functional space.

2.2. The discrete-time setting

Throughout this article, the bounded interval [0, T] is discretized into $M\in \mathbb{N}^*$ equidistant intervals $(t_i,t_{i+1}]$ of length $\Delta$ , where $\Delta=T/M$ . For a given $t\in [0,T]$ , we define $ (t)_ \Delta = \lfloor t/\Delta \rfloor \Delta $ to be its projection on the time grid. We also define $n_t=\lfloor t/\Delta \rfloor.$

For a $k\leq M$ , we set $h_k=h(k\Delta)$ . We omit the dependence on $\Delta$ to avoid cumbersome notation. Before defining the discrete-time marked Hawkes risk, we give the following stability assumption.

Assumption 2. Assume that

\begin{align*}\rho_{h,\Delta}\;:\!=\;L \sum_{k=1}^M |h_k| \Delta \mathbb{E} [b(Y)]<1,\end{align*}

where L is the Lipschitz coefficient of $\psi$ and Y is a random variable of distribution $ \nu$ . We also assume that $\nu$ has a finite first moment; that is, $\mathbb{E} [|Y|] <+\infty$ .

Just like the continuous-time case, we build the discrete-time marked Hawkes risk using the same tridimensional Poisson measure P.

Define the measurable (with respect to the filtration $(\mathcal F_{n\Delta})_{n=0,\ldots,M}$ ) sequence $(X^\Delta_k)_{k=0,\ldots,M}$ and the predictable sequence $(l^\Delta_k)_{k=0,\ldots,M}$ recursively:

(4) \begin{equation} \begin{cases} X^\Delta_0&=0, \;\; l^\Delta_0=l^\Delta_1=\psi(0), \;\; D^{\Delta}_0=0,\\[5pt] X^\Delta_n&=\int_{((n-1)\Delta,n\Delta]\times \mathbb{R}_+ \times \mathbb{R}} b(y) \mathbf 1_{\theta \leq l^\Delta_n} P (\textrm{d} s , \textrm{d} \theta, \textrm{d} y ),\\[5pt] l^\Delta_n&=\psi \left (\sum _{k=1}^{n-1} h_{n-k}X^\Delta_k\right )\!,\\[5pt] D^{\Delta}_n&= \int_{((n-1)\Delta,n\Delta]\times \mathbb{R}_+ \times \mathbb{R}} \mathbf 1_{\theta \leq l^\Delta_n} P (\textrm{d} s , \textrm{d} \theta, \textrm{d} y ), \end{cases} \text{for }n \ge 1. \end{equation}

The discrete-time Hawkes risk $R^\Delta$ and intensity $\lambda^\Delta$ are the càdlàg piecewise constant processes defined as

\begin{equation*} \begin{cases} \lambda^\Delta_t&=\lambda^\Delta_{(t)_\Delta}=l^\Delta_{n_t},\\[5pt] R^\Delta_t&=R^\Delta_{(t)_\Delta}=\sum_{k=1}^{n_t}\int_{((k-1)\Delta,k\Delta]\times \mathbb{R}_+ \times \mathbb{R}} y \mathbf 1_{\theta \leq \lambda^\Delta_{k\Delta}}P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y). \end{cases} \end{equation*}

Furthermore, the discrete-time auxiliary process can be defined as

\begin{align*}\xi^\Delta_t=\xi^\Delta_{(t)_\Delta}=\sum_{k=1}^{n_t}X_k,\end{align*}

and the discrete Hawkes process can be defined as

\begin{align*}N^{\Delta}_t=N^\Delta_{(t)_\Delta}=\sum_{k=1}^{n_t}D_k.\end{align*}

Note that despite their names, the discrete-time processes defined above are continuous-time embeddings of the times series $l^\Delta$ and $X^\Delta$ ; however, their values are allowed to change exclusively on the points of the discretization grid.

2.3. Simulation of nonlinear Poisson autoregressions

For simulation purposes, it is possible to build the sequences $X^\Delta$ and $l^\Delta$ without simulating the underlying Poisson measure P. This is based on the following observation: knowing $X^\Delta_1, \ldots, X^\Delta_n$ (and thus $l^\Delta_{n+1}$ according to (4)), the variable

\begin{align*}X^\Delta_{n+1}=\int_{(n\Delta,(n+1)\Delta]\times \mathbb{R}_+ \times \mathbb{R}} b(y) \mathbf 1_{\theta \leq l^\Delta_{n+1}} P (\textrm{d} s , \textrm{d} \theta, \textrm{d} y )\end{align*}

is a compound Poisson variable; that is,

\begin{align*}X^\Delta_{n+1}=\sum_{k=1}^{D^\Delta_{n+1}}b(Y_k^{n+1}),\end{align*}

where $D_{n+1}^\Delta | (X_1^\Delta,\ldots,X_n^\Delta) \sim \mathcal P (\Delta \cdot l^\Delta_{n+1})$ and $Y_1^{n+1},\ldots,Y_{D^\Delta_{n+1}}^{n+1}$ are i.i.d variables of distribution $\nu$ and independent of $D^\Delta_{n+1}$ .

Algorithm 1 yields the simulation of $(X_{n+1}^\Delta,R^\Delta_{(n+1)\Delta})$ given $(X_{k}^\Delta,R^\Delta_{k\Delta})_{k=1,\dots,n}$ .

Algorithm 1

Nonlinear Poisson autoregression with general kernel

Intuitively, the variable $X^\Delta_{n+1}=\sum_{k=1}^{D_{n+1}^\Delta}b(Y_k^{n+1})$ is the discrete equivalent of the increment $\textrm{d} \xi_t$ given in Definition 1. This is why we see $R^\Delta$ as a good approximation of R, which is what we will prove in the rest of this article.

We conclude this subsection by discussing the numerical cost of simulating the discrete-time process. Generally, the computation of the recursion (4) is of order $O(M^2)$ . This cost can be reduced in two cases:

1. (i) If the kernel is an Erlang function, that is, $h(t)=Q(t)e^{-\beta t}$ , where $\beta >0$ and Q is a polynomial of degree q, then the intensity is a Markov chain (up to the introduction of auxiliary processes; see [Reference Huang and Khabou23]) and the computation cost is of order O(qM).

2. (ii) If the kernel h is of compact support S, that is, $h(t)=0$ for any $ t \geq S$ , then the cost is of order O(rM), where $r=S/\Delta$ .

As an illustration, we simulate a nonlinear self-inhibiting Poisson autoregression (see Figure 1). For simplicity, we let $\nu(\textrm d y)=\delta_1(\textrm d y)$ . In this case, $X^\Delta_n=D^\Delta_n$ for all $n\ge 0$ . We choose the kernel to be

\begin{align*}h(t)=-\frac{1}{1+t^2},\end{align*}

and the jump rate $\psi(x)=(1+x)_+$ . We also simulate a discrete Poisson process of constant discrete intensity $\Delta$ (see Figure 1).

Figure 1.

Top: Discrete Poisson process of a constant deterministic intensity. Because of the independence, some nonzero counts are close to other nonzero counts. Bottom: Self-inhibiting Poisson autoregression. The realization of a nonzero count decreases the likelihood of observing counts in the near future.

Before we show that $R^\Delta$ converges to R as the time step $\Delta$ goes to zero, we discuss the choice of the spaces in which the convergence occurs.

2.4. Discussion of functional spaces

The quality of the strong approximation of continuous stochastic processes $(Z_t)_{t\in [0,T]}$ (and, in particular, diffusion SDEs) is evaluated by controlling the average uniform error

\begin{align*}\mathbb{E} \left [\sup_{0\leq t \leq T}|Z_t-Z^\Delta_t| \right]\!.\end{align*}

While the supremum norm is adapted to processes that have continuous trajectories, it can also be extended to jump-diffusion SDEs driven by a homogeneous Poisson process [Reference Bruti-Liberati and Platen7]. This is the case because the arrival times of a Poisson process $(\tau_i)$ are explicitly known and can be simulated exactly (unlike for general Hawkes processes), therefore rendering the problem equivalent to approximating a diffusion on $(\tau_i, \tau_{i+1}]$ .

The supremum norm, however, should not be used to evaluate the proximity of càdlàg trajectories.

By construction, the Hawkes process and the discrete Hawkes process are càdlàg piecewise constant functions. The Hawkes process jump times are contained in the set of the jump times of the underlying Poisson random measure. On the other hand, the jumps of the discrete Hawkes process are contained in $\{t_k,\;\;k=1,...,N\}$ , the set of the points of the subdivision. Thus, almost surely $\sup_{s\in [0,T]} | N_s - N^{\Delta}_s|{\geq} 1$ on a set with non-null probability. The processes $(N^{\Delta})_{\Delta \in (0,1)}$ will not converge to N in the uniform norm on compact sets of ${\mathbb R}^+$ almost surely. This leads us to compute the distance between the Hawkes process and the discrete Hawkes process in ${\mathbb D}([0,T],{\mathbb R})$ equipped with the Skorokhod metric

(5) \begin{equation}d_S(f,g)=\inf_{\mu \in \Lambda} \bigg \{ \sup_{0\leq t \leq T} |t-\mu(t)| \vee \sup _{0\leq t \leq T} |f(t)-g(\mu(t))| \bigg \},\end{equation}

where $\Lambda$ is the set of strictly increasing continuous functions from [0, T] to itself such that $\mu(0)=0$ and $\mu(T)=T$ . Note that since this distance allows some flexibility in the time at which the jump occurs (the role of the time change $\mu$ ), the problem that occurs with the uniform distance disappears.

By working in ${\mathbb D}([0,T],{\mathbb R})$ , we do not pay attention to the properties or the potential regularity of the paths of the processes. As increasing processes, the sample path of the Hawkes process and the discrete Hawkes process belong to a set of functions with finite bounded variation starting from 0 endowed with the distance

\begin{align*} d_{FV,T}(f,g)= \sup_{(s_i)}\sum_{i}|f(s_i) - g(s_i)|,\end{align*}

where the supremum runs over all finite partitions $(s_i)$ of $[0,T].$ It should be noticed that $\sup_{s\in [0,T]}|f(s) -g(s)| \leq d_{FV,T}(f,g)$ if $f(0)=g(0)=0.$ Thus, $N^{\Delta}$ will not converge to N when $\Delta$ converges to 0 for the $ d_{FV,T}$ distance.

As pointed out in [Reference Coutin and Decreusefond8] or [Reference Coutin, Decreusefond and Huang9], fractional Sobolev spaces provide a good framework for the study of the rate of convergence of càdlàg piecewise constant processes. According to Definition 2.1 in [Reference Bergounioux, Leaci, Nardi and Tomarelli4], for $p\geq 1$ and $0<\eta < 1$

(6) \begin{align} W_T^{\eta,p} =\{u\in L^p([0,T]),\;\; \frac{u(s)-u(t)}{(t-s)^{\eta+\frac{1}{p}}} \in L^p([0,T]^2)\}.\end{align}

They are Banach spaces endowed with the norm $\|\cdot\|_{W_T^{\eta,p}}$ , where

\begin{align*} \|u\|_{W_T^{\eta,p}}^p\;:\!=\; \int_0^T |u(s)|^p \textrm{d} s + \int \int _{[0,T] ^2} \frac{|u(s)-u(t)|^p}{(t-s)^{\eta p+1}}\textrm{d} s \textrm{d} t.\end{align*}

The classical Sobolev space is $W^{1,1}([0,T])= \{u \in L^{1}([0,T]),\;\;u(\!\cdot\!)= u(0) + \int_0^{\cdot} u'(s) \textrm{d} s \mbox{ with } u'\in L^{1}([0,T])\} $ with $\|u\|_{W^{1,1}([0,T])}= \int_0^T [|u(s)| +|u'(s)|] \textrm{d} s. $ For $p=1$ , fractional Sobolev spaces are interpolated spaces between $L^1([0,T])$ and the classical Sobolev space:

\begin{align*} L^1([0,T]) \subset W^{\eta,1}_T \subset W^{1,1}([0,T]) \quad \mbox{for all } \eta\in (0,1]\end{align*}

with continuous injections; see [Reference Bergounioux, Leaci, Nardi and Tomarelli4] for more details. One can go further. The sample paths of $R-R^{\Delta}$ are not absolutely continuous with respect to the Lebesgue measure, but they can be seen as fractional integral in the following senses. Since $R-R^{\Delta}$ is a linear combination of indicator functions, according to Example 4.1 in [Reference Bergounioux, Leaci, Nardi and Tomarelli4], there exists a process $g^{\Delta} $ with sample paths in $ L^1([0,T])$ almost surely such that $R-R^{\Delta}= I_{0^+}^{\eta}(g^{\Delta}).$ Here, for $u\in L^1([0,T]),$ the Liouville fractional of u of order $\eta$ is defined by $I_{0^+}^{\eta}(u)(t)= \frac{1}{\Gamma(\eta)}\int_0^t (t-s)^{\eta-1} u(s) \textrm{d} s,\;\;t\in [0,T].$ The Riemann–Liouville space

\begin{align*} I_{0^+}^{\eta}(L^1([0,T]))=\{u =I_{0^+}^{\eta} (u'),\;\;u' \in L^1([0,T])\}\end{align*}

endowed with

\begin{align*}\|u\|_{RL,\eta,T}= \|u'\|_{L^1([0,T])}\end{align*}

is a Banach space. According to Theorem 4.1 in [Reference Bergounioux, Leaci, Nardi and Tomarelli4], since $R-R^{\Delta}$ is piecewise constant, $\lim_{\eta \rightarrow 1} \|R-R^{\Delta}\|_{RL,\eta,T}=d_{FV,T}( R,R^{\Delta}).$ Moreover, according to Theorem 3.2 in [Reference Bergounioux, Leaci, Nardi and Tomarelli4], for $\eta',\eta \in (0,1)\;\;\eta>\eta', $

\begin{align*} W^{\eta,1}_T\cap I_{0^+}^{\eta'}(L^1([0,T])) \subset W^{\eta',1}_T,\end{align*}

with a continuous injection. Thus, we compute the distance between the Hawkes process and the discrete Hawkes process in $W^{\eta,1}([0,T])$ for all $0<\eta <1$ and in Riemann–Liouville fractional Sobolev spaces. See [Reference Bergounioux, Leaci, Nardi and Tomarelli4] for some definitions. We expect that the rate of convergence in $W^{\eta,1}_T$ will vanish when $\eta$ goes to 1.

3.1. Preliminary results

First, we give a regularity result on the limit process; that is, the continuous-time process. This is motivated by the fact that if $\lambda$ is too irregular, a piecewise approximation of it would accept (or miss) many points that are not accepted (or are rejected) in the continuous-time process.

Lemma 1. Let $h, \psi$ , and b be three functions satisfying Assumption 1. There exists a constant K such that for any $v\in [0,T]$ we have

\begin{align*}\mathbb{E} \left [|\lambda_v-\lambda_{(v)_\Delta}|\right]\leq K \left( \int_0^\Delta|h(y)|\textrm{d} y + \sup_{\epsilon \in[0, \Delta]}\int_0^{T-\Delta}\left|h(y+\epsilon)-h(y)\right|\textrm{d} y\right)\!.\end{align*}

Proof. For $v=T,$ $\lambda_v-\lambda_{(v)_\Delta}=0$ ; thus we choose $v\in [0,T).$ Since $\psi$ is L-Lipschitz and using a linear change of variables, we have

\begin{align*} \mathbb{E} \big[|\lambda_v&-\lambda_{(v)_\Delta}|\big]\\[5pt] &\leq L \mathbb{E}\left[\left | \int_0^v h(v-s)\textrm{d} \xi_s - \int_0^{(v)_\Delta}h((v)_\Delta-s)\textrm{d} \xi_s\right|\right]\\[5pt] &\leq L\left( \mathbb{E}\left[\left |\int_{(v)_\Delta}^v h(v-s) \textrm{d} \xi_s \right|\right]+ \mathbb{E} \left[\left| \int_0^{(v)_\Delta}h(v-s)-h((v)_\Delta-s) \textrm{d} \xi_s\right|\right]\right)\\[5pt] &\leq L\left( \int_{(v)_\Delta}^v \left | h(v-s)\right| \mathbb{E} [b(Y)]\mathbb{E} \left[\lambda_s\right]\textrm{d} s + \int_0^{(v)_\Delta}\left|h(v-s)-h((v)_\Delta-s)\right| \mathbb{E} [b(Y)]\mathbb{E}[\lambda_s]\textrm{d} s\!\right)\!,\end{align*}

because $\textrm{d}\mathbb{E} [ \xi_s]= \mathbb{E}[ b(Y) ]\mathbb{E}[ \lambda_s] \textrm{d} s$ . Using the bound on the expected value proved in Lemma 5, we have

\begin{align*}\mathbb{E}\left[ |\lambda_v-\lambda_{(v)_\Delta}|\right] \leq \frac{\mathbb{E} [b(Y)]L\psi(0)}{1-\mathbb{E} [b(Y)]L\|h\|_1}\left( \int_0^\Delta|h(y)|\textrm{d} y + \int_0^{(v)_\Delta}\left|h(y+v-(v)_\Delta)-h(y)\right|\textrm{d} y\right )\!,\end{align*}

and the result follows immediately.

This lemma shows that the kernel’s regularity in the sense of the shift operator in the $L^1$ norm yields the intensity’s regularity.

Since the risk process is the result of accepting the points of the underlying Poisson measure under the intensity’s curve, the approximation of the intensity is an important step in proving the convergence of $R^\Delta$ to R. We now give the first bound on the distance between intensities on the points of the discretization grid.

The following constants will be useful for the sequel.

Definition 3. We recall that $\rho_h=L\|h\|_1\mathbb{E} [b (Y)]$ and $\rho_{h,\Delta}=L\sum_{k=1}^M |h_k|\Delta \mathbb{E} [b(Y)]$ . We set

\begin{align*} C_S(h,\Delta) &= \frac{1}{(1-\rho_h)}+\frac{1}{(1-\rho_{h,\Delta})},\\[5pt] C_R(h,\Delta)&= \int_0^\Delta|h(y)|\textrm{d} y+\! \sup_{\epsilon \in [0,\Delta]}\int_0^{T-\Delta} \!|h(y+\epsilon)-h(y)|\textrm{d} y+\int_0^{T-\Delta}\! \left | h(y)-h\!\left( (y)_\Delta+\Delta\right) \right|\! \textrm{d} y. \end{align*}

The constant $C_S(h,\Delta)$ is related to the stability assumptions given by Assumptions 1 and 2, while the constant $C_R(h,\Delta)$ depends on the regularity of the kernel.

Lemma 2. Let $\lambda$ (or $\lambda^\Delta$ ) be the intensities defined by thinning from the Poisson measure P in Definition 1 (or Definition 2). Assume that Assumptions 1 and 2 hold. There exists a constant K such that for all $u\in [0,T]$ we have

(7) \begin{equation} \mathbb{E} \left[ \left| \lambda_{(u)_\Delta}- \lambda^\Delta_u \right|\right] \leq K {C_R(h,\Delta)C_S(h,\Delta)}. \end{equation}

The proof of this Lemma is given in Appendix A.

Before proving the convergence of the discrete-time Hawkes risk, we point out that if h satisfies Assumption 1 and is sufficiently regular, that is, it satisfies Assumption 3, then Assumption 2 becomes superfluous.

Assumption 3.

\begin{align*}\lim_{\Delta \to 0} \int_0^{T-\Delta} |h(t)-h((t)_\Delta+\Delta)| \textrm{d} t =0.\end{align*}

Lemma 3. If Assumptions 1 and 3 are satisfied by a kernel h, then for any $\epsilon >0$ small enough, there exists a threshold $\Delta_1 >0$ such that

\begin{align*}\frac{1}{1-\rho_{h,\Delta}} \leq \frac{1}{1-\rho_h} +\epsilon\end{align*}

for any $\Delta \leq \Delta_1$ . In particular, if there exists $\Delta_0 >0$ , Assumption 2 is verified by h for any $\Delta\leq \Delta_0$ .

Proof. For a given kernel $h \in L^1([0,T])$ we have

(8) \begin{align} \Delta\sum_{i=1}^{M-1} |h(i\Delta)| &\leq \sum_{i=1}^{M-1} \Delta\left|h(i\Delta) -\Delta^{-1} \int_{(i-1)\Delta }^{i\Delta} h(s) \textrm{d} s \right| + \sum_{i=1}^{M-1} \left| \int_{(i-1)\Delta }^{i\Delta} h(s) \textrm{d} s \right| \nonumber\\[5pt] &\leq \sum_{i=1}^{M-1} \int_{(i-1)\Delta}^{i\Delta}| h( i\Delta) -h(s)| \textrm{d} s + \int_0^T|h(s)| \textrm{d} s \nonumber \\[5pt] &= \int_0^{T-\Delta} | h(s) - h( (s)_{\Delta} +\Delta)| \textrm{d} s + \|h\|_1 \label {ineq: hyp_superflue}. \end{align}

Since the first time on the right-hand side of the last inequality tends to zero, we conclude that Assumption 2 is satisfied below a certain threshold $\Delta_1 >0$ .

To conclude this subsection, we comment on the role of the stability assumption given by Assumption 1. First, we point out that in the linear setting (that is, $\psi(x)=\mu+x$ for some $\mu>0$ ), the violation of that assumption means that $\mathbb E \left[\lambda_t\right]$ diverges. In particular, if $h(t)=\alpha\textrm e^{-\beta t}$ with $\alpha>\beta$ , then $\mathbb{E} [\lambda_t]\sim C\textrm e^{(\alpha-\beta)t}$ ; see [Reference Errais, Giesecke and Goldberg13]. We expect that the bounds proven in Lemma 1 no longer hold uniformly in time, but instead hold with a factor $\textrm e^{cT}$ for some $c>0$ , where T is the time horizon. Such results can be proven with use of the Grönwall lemma, but this is beyond the scope of this article.

As an illustration, we simulate a slightly unstable linear Hawkes process with an exponential kernel (not marked) and its discrete-time approximation in Figure 3.

Figure 3.

A realization of the discrete-time and continuous-time intensities as thinning from the same underlying Poisson measure P. The jump rate $\psi(x)=1+x$ and the kernel function $h(t)=1.01\textrm e^{-t}$ . This figure should be contrasted with Figure 2. (a) When the discretization step $\Delta$ is relatively large, instability means that the continuous-time intensity and its discrete-time approximation diverge greatly; (b) As the discretization step $\Delta$ becomes smaller, the two trajectories become closer, at least for short times. As time increases, this becomes less true as instability amplifies small differences.

With these results stated, we are now ready to give the first approximation result for the risk process. We will skip the dependence on $C_S(h,\Delta)$ in the convergence results.

3.2. Convergence in the fractional Sobolev space

The goal of this subsection is to give a bound on the distance between the risk process on the compact interval [0, T] and its discrete counterpart in the fractional Sobolev space defined in (6). Recall that for $\eta\in(0,1)$ , $W^{\eta,q}_T\;:\!=\; \left\{ u \in L^q ([0,T]): \|u\|_{W^{\eta,q}_T} <+\infty\right \}.$ Here we restrict ourselves to $q=1$ and omit it from the notation. We also recall the fractional integral. For $u\in L^1([0,T])$ and $0<\eta<1$ ,

\begin{align*} I_{0^+}^{\eta} (u)(t)= \frac{1}{\Gamma(\eta)}\int_0^t (t-s)^{\eta- 1} u(s) \textrm d s,\;\;t\in [0,T],\end{align*}

and the Riemann-Liouville space

\begin{align*} I_{0^+}^{\eta}(L^1([0,T]))=\{u \in L^1([0,T]) \left|\right. \mbox{there exists } v \in L^1([0,T]),\;\;u=I_{0^+}^{\eta}(v) \}. \end{align*}

Proposition 1. Let $T>0.$ Let $( h, \psi, \nu,b)$ fulfill Assumption 1. Let R (or $R^\Delta$ ) be the continuous-time (or discrete-time) risk process. We have

\begin{align*}(R_t)_{t\in [0,T]} \in W^\eta_T\cap I_{0^+}^{\eta}(L^1([0,T]))\textit{ and }(R^{\Delta}_t)_{t\in [0,T]} \in W^\eta_T\cap I_{0^+}^{\eta}(L^1([0,T])) \end{align*}

almost surely.

Proof. Note that $R_t= \sum_{k=1}^{N_t} Y_k$ and $R_t^{\Delta}= \sum_{k=1}^{N_t^{\Delta}} Y_k \mbox{ for all } t\in [0,T].$ Moreover, N and $N^{\Delta}$ are counting processes with finite expectation. Indeed,

\begin{align*}{\mathbb E}[ N_T]+ {\mathbb E}[ N_T^{\Delta}]=\int_0^T \left( {\mathbb E}[\lambda_s] +{\mathbb E}[\lambda_s^{\Delta}]\right) \textrm{d} s <\infty, \end{align*}

according to Lemmas 5 and 4. Thus, the processes R and $R^{\Delta}$ are linear combination of indicators of finite subinterval of the form [a, T] with $a>0$ of [0, T] which belongs to $W^{\eta}_T\cap I_{0^+}^{\eta}(L^1([0,T]))$ according to Example 4.1 in [Reference Bergounioux, Leaci, Nardi and Tomarelli4].

We now give a bound on the difference between the aggregate risk observed for the continuous-time risk process and its discrete-time counterpart, which will be crucial in proving convergence in the fractional Sobolev space.

Proposition 2. Let $T>\Delta>0.$ Let $(h, \psi,\nu,b)$ fulfill Assumptions 1 and 2. Let R (or $R^\Delta$ ) be a continuous-time (or discrete-time) Hawkes risk process, defined by thinning from the same underlying Poisson measure P.

There exists a constant K such that for all $0\leq s \leq t \leq T$

\begin{align*} \mathbb{E} \left [|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right] \leq K \left(C_R(h,\Delta) \left[ (t)_\Delta-(s)_\Delta\right] + \Delta \right )\!, \end{align*}

where $C_R(h,\Delta) $ is given in Definition 3.

The proof of this proposition is given in Appendix A. Before we move to strong functional convergence results, we give in Figure 4 a numerical illustration of Proposition 2 for the simple Hawkes process of kernel $h(t)=\frac{0.6\cdot \cos(t)}{1+t^2}$ .

Figure 4.

A Monte Carlo approximation of $\mathbb{E} \left[|N_T-N^{\Delta}_T|\right]$ for $T=5$ (blue), and the least-squares linear approximation (orange), with the equation being $y= 8.4 \cdot \Delta^{1.1}$ .

We are now ready to give the first general strong convergence result for the discrete-time Hawkes risk processes.

Theorem 1. Let $T>0.$ Let $(h, \psi,\nu,b)$ fulfill Assumptions 1 and 2. Let R (or $R^\Delta$ ) be a continuous-time (or discrete-time) Hawkes risk process. We have

\begin{align*}\mathbb{E} \left[\|R-R^\Delta\|_{W^\eta_T}\right] \leq K\left[T^{2} C_R(h,\Delta) + T\Delta^{1-\eta}\right ]\!,\end{align*}

where $C_R(h,\Delta)$ is defined in Definition 3 and K is a positive multiplicative constant depending on $\eta$ that does not depend on T or $\Delta$ .

The strong approximation result in Theorem 1 is not useful for numerical approximations, because the dependence of $C_R(h,\Delta)$ on $\Delta$ is not explicit a priori. It turns out that if the kernel is of finite p-variations, we can give the order of convergence in $\Delta$ .

For $p\geq 1$ , the p-variation of a function $f:[0,T]\to \mathbb{R}$ is defined as

\begin{align*}\|f\|_{p\text{-var}}\;:\!=\;\left(\sup _{ \mathcal D} \sum_{t_i\in\mathcal D} |f(t_{i+1})-f(t_i)|^p\right)^{1/p},\end{align*}

where $\mathcal D$ is the set of finite subdivisions of [0, T].

The set of functions of finite p-variation contains piecewise Hölder functions of a Hölder index $p^{-1} \in(0,1]$ . As we will see in Corollary 1, these functions are regular enough to have more explicit rates of convergence. Using Lemma 9, we derive Corollary 1.

Corollary 1. Let $T> 0$ and $p\geq 1$ and assume that Assumption 1 holds. Moreover, assume that the kernel h is of a finite p-variation $\|h\|_{p\text{-var}}$ . Then, for $\Delta$ small enough, Assumption 3 is fulfilled and

\begin{align*}C_R(h,\Delta) \leq K\Big[ \|h\|_{p\text{-var}} T^{\frac{p-1}{p}}\Delta^{\frac{1}{p}}+ \Delta \|h\|_{\infty}\Big].\end{align*}

Moreover,

\begin{align*}\mathbb{E} \big[ \|R-R^\Delta\|_{W^\eta_T}\big]\leq K_{\eta}\Big[\|h\|_{p\text{-var}}T^{\frac{3p-1}{p}}\Delta^{\frac{1}{p}}+ T \Delta^{1-\eta}+\Delta \|h\|_{\infty}\Big]\end{align*}

for some positive constant $K_{\eta}$ that does not depend on T or $\Delta$ .

We now turn to the proof of Theorem 1.

Proof. First, we recall the result of Proposition 2:

\begin{align*} \mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right] \leq &K \left[C_R(h,\Delta)\left((t)_\Delta-(s)_\Delta\right)+ \Delta\right]\!.\end{align*}

The norm of the difference between the continuous-time risk process and its discrete-time counterpart in the fractional Sobolev space is written as

\begin{align*} \mathbb{E} \big[\|R-R^\Delta\|_{W^\eta_T}\big] &=\int_0^T \mathbb{E}\left[|R_t-R^\Delta_t| \right]\textrm{d} t + \int_0^T \int_0^T \frac{\mathbb{E}\left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s. \end{align*}

We bound each of the three terms individually. We start with the second term. Thanks to the symmetry of the integrals with respect to s and t,

\begin{align*} &\int_0^T \int_0^T \frac{\mathbb{E}\left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\quad= \sum_{i,j=1}^M \int_{(i-1)\Delta}^{i\Delta\wedge T}\int_{(j-1)\Delta}^{j\Delta\wedge T} \frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\quad = \sum_{i=1}^M\int_{(i-1)\Delta}^{i\Delta\wedge T} \int_{(i-1)\Delta}^{i\Delta\wedge T} \frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\qquad +2\sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta \wedge T} \frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\qquad +2\sum_{i=1}^{M-2}\sum_{j=i+2}^M \int_{(i-1)\Delta}^{i\Delta} \int_{(j-1)\Delta}^{j\Delta\wedge T} \frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s. \end{align*}

For the first term, since s and t are in the same interval $[(i-1)\Delta, i \Delta),$ we have $R_t^\Delta- R_s^\Delta=0.$ Moreover,

\begin{align*} \mathbb{E}\left[|R_t-R_s|\right] \leq \mathbb{E} \left[\sum_{i=N_s+1}^{N_t} |Y_i|\right] = \mathbb{E}[|Y|]\mathbb{E}[N_t-N_s] = K \int_s^t \mathbb{E}[\lambda_u]\textrm{d}u. \end{align*}

Using Lemma 5 and the fact that $\rho_{h} <1$ , we have

\begin{align*} \mathbb{E} \left[|R_t-R_s|\right] &\leq \frac{K(t-s)}{1-\rho_{h}}\\[5pt] &\leq K (t-s). \end{align*}

Therefore,

\begin{align*} \sum_{i=1}^M \int_{(i-1)\Delta}^{i\Delta\wedge T}\int_{(i-1)\Delta}^{i\Delta\wedge T} \frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s &= \sum_{i=1}^M \int_{(i-1)\Delta}^{i\Delta\wedge T}\int_{(i-1)\Delta}^{i\Delta\wedge T} \frac{\mathbb{E}\left[|R_t-R_s|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\quad \leq K\sum_{i=1}^M \int_{(i-1)\Delta}^{i\Delta\wedge T}\int_{(i-1)\Delta}^{i\Delta\wedge T} \frac{|t-s|}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\quad \leq K\sum_{i=1}^M \int_{(i-1)\Delta\wedge T}^{i\Delta\wedge T}\int_{(i-1)\Delta}^{t} \frac{ \textrm{d} s}{(t-s)^{\eta }} \textrm{d} t \\[5pt] &\quad \leq K\sum_{i=1}^M \int_{(i-1)\Delta}^{i\Delta\wedge T} \left(t-(i-1)\Delta \right)^{1-\eta} \textrm{d} t \\[5pt] &\quad \leq K_{\eta} T \Delta^{1-\eta}. \end{align*}

When s and t are in adjacent bins, $R^\Delta_t-R^\Delta_s=\sum_{k=1}^{D^\Delta_{n_t}}Y_j$ in distribution, where

$D^\Delta_{n_t}= \int_{((n_t-1)\Delta, n_t \Delta]\times \mathbb{R}_+ \times \mathbb{R}} \mathbf 1_{\theta \leq \lambda_u^{\Delta}} P (\textrm{d} u, \textrm{d} \theta, \textrm{d} y).$ Thus,

\begin{align*} \sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta\wedge T} &\frac{\mathbb{E} \left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] =& \sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta\wedge T} \frac{\mathbb{E}\left[\left|R_t-R_s-\sum_{j=1}^{D^\Delta_{n_t}-1}Y_j\right|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] \leq & \sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta\wedge T} \frac{\mathbb{E}\left[|R_t-R_s| \right]+ \mathbb{E} [|Y|]\mathbb{E}\left[ D^\Delta_{n_t}\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s. \end{align*}

And since $\mathbb{E} \left[D^\Delta_{n_t} \right]= \int_{(n_t-1)\Delta}^{n_t \Delta}\mathbb{E}\left[ \lambda_u^{\Delta} \right]\textrm{d} u ,$ we bound $\lambda_u^{\Delta} $ using Lemma 4 and the definition of $C_S(h,\Delta)$ given in Definition 3:

\begin{align*} \sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta\wedge T}& \frac{\mathbb{E}\left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] & \leq K\sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \int_{i\Delta}^{(i+1)\Delta\wedge T} \frac{1}{(t-s)^\eta}+ \frac{\Delta}{(t-s)^{1+\eta}} \textrm{d} t \textrm{d} s \\[5pt] & \leq K\sum_{i=1}^{M-1}\int_{(i-1)\Delta}^{i\Delta} \Big(((i+1)\Delta-s)^{1-\eta}-(i\Delta-s)^{1-\eta}\\[5pt] &\quad - \Delta((i+1)\Delta-s)^{-\eta}+\Delta (i\Delta-s)^{-\eta} \Big)\textrm{d} s \\[5pt] &= K \sum_{i=1}^{M-1}\int_{0}^{\Delta} (x+\Delta)^{1-\eta}- x^{1-\eta}-\Delta (x+\Delta)^{-\eta}+ \Delta x^{-\eta} \textrm{d} x \\[5pt] &\leq K \sum_{i=1}^{M-1} \Delta^{2-\eta}\\[5pt] & \leq K_{\eta}T \Delta^{1-\eta}. \end{align*}

We now treat the third-case scenario, when s and t are separated by more than one bin. Keeping in mind that $\Delta <t-s$ implies $((t)_\Delta-(s)_\Delta) \leq (t-s+\Delta)\leq 2(t-s)$ and using the result of Proposition 2 and Remark 3, we have

\begin{align*} \sum_{i=1}^{M-2}\sum_{j=i+2}^M \int_{(i-1)\Delta}^{i\Delta} \int_{(j-1)\Delta}^{j\Delta\wedge T}& \frac{\mathbb{E}\left[|(R_t-R_s)-(R^\Delta_t-R^\Delta_s)|\right]}{\mathbb{E} |Y_1||t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] & \leq K \sum_{i=1}^{M-2}\sum_{j=i+2}^M \int_{(i-1)\Delta}^{i\Delta} \int_{(j-1)\Delta}^{j\Delta\wedge T} \frac{ C_R(h,\Delta)(t-s)+ \Delta }{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s \end{align*}

\begin{align*} &\leq K \sum_{i=1}^{M-2} \int_{(i-1)\Delta}^{i\Delta} \int_{(i+1)\Delta}^T \frac{C_R(h,\Delta) (t-s)+ \Delta }{|t-s|^{1+\eta }} \textrm{d} t \textrm{d} s\\[5pt] &\leq K\sum_{i=1}^{M-2} \int_{(i-1)\Delta}^{i\Delta} C_R(h,\Delta) (t-s)^{1-\eta}+ \Delta ( (i+1)\Delta -s)^{-\eta} \textrm{d} s\\[5pt] &\leq K_{\eta} \left[C_R(h,\Delta)T^{2-\eta} + T\Delta^{1-\eta}\right]\!. \end{align*}

For the first term, using the fact that $t_{\Delta} \leq t$ , we have

\begin{align*} \int_0^T {\mathbb{E} \left[|R_t-R^\Delta_t|\right]}\textrm{d} t & \leq K\int_0^T ( C_R(h,\Delta )t_{\Delta} +\Delta)\textrm{d} t \\[5pt] &\leq K\int_0^T ( C_R(\Delta) t +\Delta)\textrm{d} t \\[5pt] &\leq K \left[C_R(\Delta) T^2 + \Delta T\right]\!. \end{align*}

This yields the first explicit speed of convergence of the discrete-time marked Hawkes risk for a rich family of kernels. We point out that convergence rates can also be obtained for different kernels that do not lie in the set of functions with finite p-variations. A particularly interesting example is the Hawkes process driven by the kernel

\begin{align*}h(t)=\frac{C}{\sqrt{t}}\mathbf 1_{t\in (0,T]},\end{align*}

where $C>0$ is a constant that ensures that Assumption 1 is in force.

Since this kernel is not of finite p-variation, it should be verified that Assumption 2 holds. This is possible because the sum of the inverses of the square root can be bounded by the integral of $\frac{1}{\sqrt x}$ . Once this is done, we can apply Theorem 1. Using the fact that h is decreasing on (0, T], we can bound the modulus of continuity of the shift operator in an elementary fashion:

\begin{align*} \int_0^{T-\Delta} |h(y+\epsilon)-h(y)|\textrm{d} y &= \int_0^{T-\Delta} h(y)-h(y+\epsilon) \textrm{d} y \\[5pt] &=\int_0^{T-\Delta} h(y) \textrm{d} y - \int_0^{T-\Delta} h(y+\epsilon) \textrm{d} y \\[5pt] &=O(\Delta^{\frac{1}{2}}).\end{align*}

Hence,

\begin{align*}\mathbb{E}\left[ \|R^\Delta- R\|_{W^\eta_T} \right]\leq K (T \Delta^{\frac{1}{2}}+ T \Delta^{1-\eta}).\end{align*}

This rate is slower than the rate of convergence for Hawkes risks whose kernels are of bounded variations $K(T^2 \Delta + T \Delta^{1-\eta})$ , which is natural because of the singularity of the inverse square root near zero.

We now prove the convergence in the space of càdlàg functions equipped with the Skorokhod metric for a class of Hawkes processes.

3.3 Convergence in the Skorokhod space

We start this section by recalling the definition of the Skorokhod space. We call $\mathbb D([0,T],{\mathbb R})$ the space of right continuous functions, with left limit (càdlàg). The canonical metric over this space is the Skorokhod metric defined by (5). The fact that this distance allows for small uncertainties in time, unlike the uniform distance, ensures that it is well adapted for Hawkes risk processes whether in continuous time or in discrete time. The different properties of the Skorokhod space can be found in [Reference Billingsley5]. Let $\Lambda$ denote the class of strictly increasing continuous mapping of $|0,T]$ into itself such that $\lambda(0)=0,$ $\lambda(T)=T.$ For x and y in $\mathbb D([0,T])$ ,

\begin{align*} d_S(x,y)= \inf_{\lambda\in \Lambda} \left \{\|\lambda-\textrm{Id}\|_{\infty} \vee \|x-y\lambda \|_{\infty} \right\},\end{align*}

where Id is the identity map from $[0,T].$

Since the jump times of the continuous-time Hawkes risk R occur almost surely outside the time grid $\sigma=(k\Delta)_{k=1,\ldots,M}$ , the projection $(A_\sigma R)_{t\in [0,T]} \;:\!=\; (R_{(t)_\Delta})_{t\in [0,T]}$ represents an intermediate process between R and $R^\Delta$ ; hence,

\begin{align*}d_S(R,R^\Delta) \leq d_S(R,A_\sigma R) + d_S (A_\sigma R, R^\Delta).\end{align*}

The first term on the left-hand side simply evaluates the distance between the path of R and its projection on the grid, and is bounded (see [Reference Billingsley5], Lemma 3, p. 127) by

\begin{align*}d_S(R,A_\sigma R) \leq \Delta \vee \omega'_R(\Delta),\end{align*}

where $\omega'_R(\Delta)$ is the $\Delta$ modulus of continuity for the càdlàg process

\begin{align*}\omega'_X(\Delta)=\inf_{\Delta\text{-sparse}}\max _{1\leq i \leq M} \sup_{u,v \in [t_{i-1},t_{i})}|X_u-X_v|,\end{align*}

the infimum being taken on the set of all partitions $\{t_i\}$ of [0, T] such that $\min t_i-t_{i-1} > \Delta$ . Therefore, the Skorokhod distance between the continuous-time Hawkes risk and its discrete-time counterpart is controlled by

\begin{align*}d_S(R, R^\Delta) \leq \Delta + \omega'_R(\Delta)+ d_S (A_\sigma R, R^\Delta).\end{align*}

It is then enough to control the regularity of R (using $(\omega'_R(\Delta))$ and its distance from $R^\Delta$ on the points of the grid $(d_S (A_\sigma R, R^\Delta))$ . Indeed, noticing that both $A_\sigma R$ and $R^\Delta$ are constant on $[k\Delta, (k+1)\Delta)$ for $k=1,\ldots,M$ , we immediately see that

\begin{align*} d_S(A_\sigma R , R^\Delta) & \leq \|A_\sigma R - R^\Delta\|_\infty\\[5pt] &= \max_{k=1,\ldots,M} |R_{k\Delta}-R^\Delta_{k\Delta}|,\end{align*}

yielding

(9) \begin{equation}d_S(R,R^\Delta) \leq \Delta + \omega'_R(\Delta) + \max_{k=1,\ldots,M} |R_{k\Delta}-R^\Delta_{k\Delta}|.\end{equation}

Before giving an upper bound on the distance between the two processes evaluated on the points of the grid, we remind the reader that $\mathcal F$ is the filtration associated with the common underlying Poisson measure P.

Proposition 3. Assume that Assumptions 1 and 3 or Assumptions 1 and 2 are in force. Assume also that $\nu$ has a finite second moment. Let

\begin{align*}\Xi_k\;:\!=\; R_{k\Delta}-\mathbb{E} [Y]\int_0^{k\Delta} \lambda _s \textrm{d} s\end{align*}

and

\begin{align*}\Xi^\Delta_k \;:\!=\;R^{\Delta}_{k\Delta} - \mathbb{E} [Y] \sum _{i=1}^k \lambda^\Delta_{i\Delta} \Delta.\end{align*}

Then $(\Xi_{k})_{k=0,\ldots, M}$ and $(\Xi^\Delta_k)_{k=0,\ldots, M}$ are $(\mathcal F_{k\Delta})_{k=1,\ldots,M}$ -martingales. Moreover,

\begin{align*}\mathbb E \Big[\max_{k=1,\ldots,M} |\Xi_{k}-\Xi^\Delta_{k}|\Big] \leq K \sqrt{T C_R(h,\Delta)},\end{align*}

where K is a positive multiplicative constant that does not depend on T or $\Delta$ and $C_R(h,\Delta)$ is defined in Definition 3.

The proof of this Proposition can be found in Appendix A.

We finally give an upper bound on the Skorokhod distance between the continuous-time Hawkes risk and its discrete-time counterpart in the case that the jump rate $\psi$ is bounded.

Theorem 2. Suppose that Assumptions 1 and 3 or Assumptions 1 and 2 are in force. Assume furthermore that $\psi$ is bounded and that $\nu$ has a finite second moment.There exists a positive constant K that does not depend on T and $\Delta$ such that

\begin{align*} \mathbb{E} \left[d_S(R,R^\Delta)\right] &\leq K \left( \Delta \left(1+ \|\psi\|_{\infty}[ 1+ \|\psi\|_{\infty}^2T^2] \right)+ \sqrt{T C_R(h, \Delta) } +T C_R(h, \Delta)\right)\!. \end{align*}

Moreover, if h is of finite p-variations for $p\geq 1$ , then Assumption 3 automatically holds and

\begin{align*} \mathbb{E} \left[d_S\left(R,R^\Delta\right)\right] &\leq K \Delta \left(1+\|\psi\|_{\infty}\left(1+T^2 \|\psi\|_{\infty}^{2}\right)\right) \\[5pt] &\quad + K\left(\sqrt{ \|h\|_{p\text{-var}}T^{\frac{2p-1}{p}}\Delta^{\frac{1}{p}} + T\Delta \|h\|_{\infty} } + \|h\|_{p\text{-var}}T^{\frac{2p-1}{p}}\Delta^{\frac{1}{p}}+T\Delta \|h\|_{\infty} \right)\!. \end{align*}

Proof. Taking the expected value of the inequality (9), we have

\begin{align*} \mathbb{E} \left[d_S\left(R,R^\Delta\right)\right] &\leq \Delta + \mathbb{E} [\omega '_R (\Delta)]+ \mathbb{E} \left[\max_{k=1,\ldots,M} |R_{k\Delta}-R^\Delta_{k\Delta}|\right] \\[5pt] & \leq \Delta + \mathbb{E} [\omega '_R (\Delta)] + \mathbb{E} \left[\max_{k=1,\ldots,M} |\Xi_k-\Xi^\Delta_{k}|\right] \\ & \quad + \mathbb{E} \left[\max_{k=1,\ldots,M} \left | \int_0^{k\Delta}\lambda_s \textrm{d} s- \sum_{i=1}^k \lambda_{i\Delta}^\Delta \Delta \right|\right]\\[5pt] &= \Delta+ A_1 +A_2 +A_3. \end{align*}

The difference between the values for the discrete-time and continuous-time processes has been separated into a martingale part and a compensator part.

The martingale term $A_2$ has already been dealt with in Proposition 3. For $A_3$ , we simply write

\begin{align*} \left | \int_0^{k\Delta}\lambda_s \textrm{d} s- \sum_{i=1}^k \lambda_{i\Delta}^\Delta \Delta \right |& =\left | \sum _{i=1}^k \int_{(i-1)\Delta}^{i\Delta} \lambda_ s - \lambda^\Delta_{i\Delta} \textrm{d} s\right |\\[5pt] & \leq \sum _{i=1}^M \int_{(i-1)\Delta}^{i\Delta} |\lambda_s - \lambda_{i\Delta}^\Delta| \textrm{d} s. \end{align*}

Taking the expectation of each term of the previous inequality and using the inequality (7), we obtain

\begin{align*} A_3 \leq K T C_R(h,\Delta). \end{align*}

The last term is $A_1$ . Since the jump rate is bounded by $\|\psi\|_\infty$ , the compound Poisson process

\begin{align*}\Pi_t= \int_{(0,t]\times \mathbb{R}_+ \times \mathbb{R}}|y| \mathbf 1_{\theta \leq \|\psi\|_\infty} P (\textrm{d} s, \textrm{d} \theta, \textrm{d} y)\end{align*}

dominates the process R. That is, for any $0\leq a \leq b \leq T$ , $\Pi_b-\Pi_a \geq \left|R_b-R_a\right|.$ The problem now is to determine the behavior of the modulus of continuity of a compound Poisson process of intensity $\|\psi\|_\infty $ , which is solved in Lemma 8.

The case of a bounded jump rate is quite restrictive; for instance, the results of Theorem 2 cannot be applied to the standard linear Hawkes process. We therefore give a generalization to unbounded jump rates in the following theorem.

Theorem 3. Suppose that Assumptions 1 and 3 or Assumptions 1 and 2 are in force. Assume that h is bounded and that $\nu$ has a finite second moment. There exists a positive constant K that does not depend on T and $\Delta$ such that

\begin{align*} \mathbb{E} \left[d_S\left(R,R^\Delta\right)\right] &\leq K \left( { \Delta^{\frac{1}{4}} \left(1+ T^2\right)} + \sqrt{T C_R(h, \Delta) } +T C_R(h, \Delta)\right)\!. \end{align*}

Moreover, if h is of finite p-variations for $p\geq 1$ , then Assumption 3 automatically holds and

\begin{align*} \mathbb{E} \left[d_S(R,R^\Delta)\right] \leq &K \left( { \Delta^{\frac{1}{4}} \left(1+ T^2\right)} +T\Delta \right)\\[5pt] & +K\left( \sqrt{ \|h\|_{p\text{-var}}T^{\frac{2p-1}{p}}\Delta^{\frac{1}{p}} + T\Delta \|h\|_{\infty} } + \|h\|_{p\text{-var}}T^{\frac{2p-1}{p}}\Delta^{\frac{1}{p}} + T\Delta \|h\|_{\infty}\right)\!. \end{align*}

Proof. Let C a positive real number to be fixed later and let $\psi^C= \psi \wedge C.$ We also denote by $ \lambda^C,\;\;N^C,\;\;R^C$ the intensity, Hawkes process and risk process, and by $ \lambda^{C,\Delta},\;\;N^{C,\Delta},\;\;R^{C,\Delta}$ the discrete intensity, Hawkes process and risk process associated with $\psi^C.$ On the event $\Gamma_C=\{\omega, \sup_{t\leq T} \lambda_t \leq C\},$ the process $\left(\lambda^C_t,N^C_t,R^C_t,\;\;t\in [0,T]\right)$ and $(\lambda_t,N_t, R_t,\;\;t\in [0,T])$ coincide, and on the event $\Gamma_{C,\Delta}=\left\{\omega, \sup_{t\leq T} \lambda_t^{\Delta} \leq C\right\},$ the processes $\left(\lambda^{C,\Delta}_t,N^{C,\Delta}_t,R^{C,\Delta}_t,\;\;t\in [0,T]\right)$ and $\left(\lambda^{\Delta}_t,N_t^{\Delta}, R^{\Delta}_t,\;\;t\in [0,T]\right)$ coincide.

On the event $\Omega \setminus \left(\Gamma_C\cap \Gamma_{C,\Delta}\right)\!,$ we estimate

\begin{align*} d_S\left(R,R^{\Delta}\right) \leq \sup_{s\leq T}|R_s| + \sup_{s\leq T} |R_s^{\Delta}| \leq \sum_{k=1}^{N_T}|Y_k| + \sum_{k=1}^{N_T^{\Delta}}|Y_k|.\end{align*}

Thus,

\begin{align*} \mathbb{E} \left[d_S\left(R,R^{\Delta}\right)\right] &= \mathbb{E} \left[d_S\left(R,R^{\Delta}\right)\mathbf 1_{\Gamma_C \cap \Gamma_{C,\Delta}}\right] + \mathbb{E} \left[d_S\left(R,R^{\Delta}\right) \left(\mathbf 1_{\Gamma_C^c \cup \Gamma_{C,\Delta}^c}\right)\right]\\[5pt] &\leq \mathbb{E} \left[d_S\left(R^C,R^{C,\Delta}\right)\right]+ \mathbb{E} \left[d_S\left(R,R^{\Delta}\right) \left(\mathbf 1_{\Gamma_C^c } + \mathbf 1_{ \Gamma_{C,\Delta}^c}\right)\right]\!.\end{align*}

Using Cauchy-Schwarz and Markov inequalities, we obtain

\begin{align*} &\mathbb{E} \big[d_S\big(R, R^{\Delta} \big)\big]\\[5pt] & \quad \leq \mathbb{E} \left[d_S\big( R^C,R^{C,\Delta }\big)\right] + \sqrt{{4} \mathbb{E} \left(\left[\sum_{k=1}^{N_T}|Y_k|\right]^2 +\left[\sum_{k=1}^{N_T^{\Delta}}|Y_k|\right]^2\right) \left({\mathbb P} \left(\Gamma_C^c \right) +{\mathbb P} \left(\Gamma_{C,\Delta}^c \right) \right)}\\[5pt] & \quad \leq \mathbb{E} \left[d_S\big( R^C,R^{C,\Delta }\big)\right] + \frac{2}{C} \sqrt{ \mathbb{E} \left(\left[\sum_{k=1}^{N_T}|Y_k|\right]^2 +\left[\sum_{k=1}^{N_T^{\Delta}}|Y_k|\right]^2\right) \mathbb{E} \left(\sup_{s \leq T} \lambda_s^2 + \sup_{s\leq T}(\lambda_s^{\Delta})^2\right)} .\end{align*}

We now bound each term under the square root.

First, since $N_t =\int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} \mathbf 1_{\theta \leq \lambda _s} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y)$ ,

\begin{align*}\mathbb{E} [ N_T^2] &\leq 2 \int_0^T \mathbb{E}[\lambda_s ]\textrm{d} s + 2\mathbb{E}\left[\int_0^T \lambda_s \textrm{d} s \right]^2\\[5pt] &\leq 2T \sup_{s\leq T} \mathbb{E}[\lambda_s ] + 2T^2\sup_{s\leq T} \mathbb{E}[\lambda_s^2 ].\end{align*}

According to Lemmas 5 and 7,

(10) \begin{align} \mathbb{E} [ N_T^2] \leq K\big(1+T^2\big).\end{align}

Second, since $N_t^{\Delta} =\int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} \mathbf 1_{\theta \leq \lambda _s^{\Delta}} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y)$ ,

\begin{align*}\mathbb{E} [ \big(N_T^{\Delta}\big)^2] &\leq 2 \int_0^T \mathbb{E}[\lambda_s ^{\Delta}]\textrm{d} s + 2\mathbb{E}\left[\int_0^T \lambda_s^{\Delta}\textrm{d} s \right]^2\\[5pt] &\leq 2T \sup_{s\leq T} \mathbb{E}[\lambda_s ^{\Delta}] + 2T^2\sup_{s\leq T} \mathbb{E}[\big(\lambda_s^{\Delta}\big)^2 ].\end{align*}

According to Lemmas 4 and 6,

\begin{align*} \mathbb{E} [\big(N_T^\Delta\big)^2] \leq K\big(1+T^2\big).\end{align*}

Third, recall that

\begin{align*} \lambda_t = \psi \left( \int_{0}^{t-} h(t-s) \textrm{d} \xi_s\right)\!.\end{align*}

Thus, since $\psi$ is Lipschitz continuous,

\begin{align*} \lambda_s &\leq \psi(0) +L \int_{0}^{t-} h(t-s) \textrm{d} \xi_s.\end{align*}

Since $\xi_t= \int_{(0,t] \times \mathbb{R}_+ \times \mathbb{R}} b(y)\mathbf 1_{\theta \leq \lambda _s} P(\textrm{d} s, \textrm{d} \theta, \textrm{d} y)$ is an increasing process and h is bounded,

\begin{align*} \lambda_s &\leq \psi(0) +L \|h\|_{\infty} \xi_T.\end{align*}

Thus,

\begin{align*} \sup_{s\leq T} \lambda_s^2 &\leq 2 \psi(0) + 2L^2 \|h\|_{\infty}^2 \xi_T^2.\end{align*}

Then, taking the expectation of each term, we obtain

\begin{align*} \mathbb{E} \bigg[\sup_{s \leq T} \lambda_s^2\bigg] \leq 2 \psi(0) + 2L^2 \|h\|_{\infty}^2 \mathbb{E} \big[\xi_T^2\big]. \end{align*}

Using the same line as in the proof of the estimation of $\mathbb{E}[N_T^2],$ (10), we have

(11) \begin{align} \mathbb{E} \bigg[\sup_{s \leq T} \lambda_s^2\bigg] \leq K\big(1+T^2\big). \end{align}

Fourth, recall that $\lambda_t^{\Delta}= \psi \left(\sum_{k=1}^{n_t-1}h(n_t-k) X_k^{\Delta}\right)\!. $ Since $\psi$ is Lipschitz continuous,

\begin{align*} \lambda_t^{\Delta} \leq \psi(0) + L\sum_{k=1}^{n_t-1} h(n_t-k) X_k^{\Delta}. \end{align*}

Using the fact that h is bounded and

\begin{align*}X_n^{\Delta} =\int_{((n-1)\Delta,n\Delta]\times \mathbb{R}_+ \times \mathbb{R}} b(y) \mathbf 1_{\theta \leq l^\Delta_n} P (\textrm{d} s , \textrm{d} \theta, \textrm{d} y ),\end{align*}

we have

\begin{align*} \sup_{s\leq T} \lambda_s^{\Delta} \leq \psi(0) + L\|h\|_{\infty}\int_{(0,T]\times \mathbb{R}_+ \times \mathbb{R}} b(y) \mathbf 1_{\theta \leq \lambda^\Delta_s} P (\textrm{d} s , \textrm{d} \theta, \textrm{d} y ). \end{align*}

Using the same lines as in the proof of the estimation (11), we obtain

\begin{align*} \mathbb{E} \left[\sup_{s \leq T} \left(\lambda_s^{\Delta}\right)^2\right] \leq K\left(1+T^2\right)\!. \end{align*}

Then for a universal constant K,

\begin{align*} \mathbb{E} \left[d_S\big(R,R^{\Delta}\big)\right] &\leq \mathbb{E} \left[d_S\big( R^C,R^{C,\Delta }\big)\right] + K \frac{ 1+T^2}{C}.\end{align*}

The quantity $\mathbb{E}\left[d_S\big( R^C,R^{C,\Delta}\big)\right]$ is bounded with the use of Theorem 2 for $\psi^C$ with $\|\psi^C\|_{\infty}\leq C,$

\begin{align*} \mathbb{E} \left[d_S\big(R,R^{\Delta}\big)\right] &\leq K \left( \Delta {\left[ 1+ C\big(1+C^2T^2\big)\right]} + \sqrt{T C_R(h, \Delta) } +T C_R(h, \Delta)\right)+ K \frac{ 1+T^2}{C}.\end{align*}

Taking $C={\frac{1}{\Delta^{\frac{1}{4}}}}$ , we derive

\begin{align*} \mathbb{E} \left[d_S(R,R^{\Delta})\right] &\leq K { \Delta^{\frac{1}{4}} \big(1+ T^2\big)} +K\left( \sqrt{T C_R(h, \Delta) } +T C_R(h, \Delta)\right )\!,\end{align*}

where K is a constant independent of T and $\Delta.$

Remark 5. If in the proof of Theorem 3, we bound ${\mathbb P}(\Gamma_C^c)$ by $\frac{{\mathbb E}[ \sup_{s\leq T} \lambda_s]}{C}$ instead of $\frac{{\mathbb E}[ \sup_{s\leq T} \lambda_s^2]}{C^2}$ , we obtain

\begin{align*} \mathbb{E} \left[d_S \big(R,R^{C,\Delta}\big) \right]& \leq K \left ( {\Delta^{\frac{1}{7}}\big(1+T^{\frac{11}{7}}}\big) + \sqrt{T C_R(h,\Delta)} + T C_R(h,\Delta)\right)\!.\end{align*}

The power of $T$ in the constant is smaller than in the bound given in Theorem 3 but the rate of convergence in $\Delta$ is slower.