Concerns about the definition of Hawkes process

Question

In the lecture notes I am reading about Point process, when we introduced the Hawkes process several expressions are given and I have some difficulty to understand properly what is the $Z_t$ (defined below). I tried to make my doubts clear by precising two points w of concerns.

Before I introduce the framework in order to fix notations.

We consider $T_1<…<T_n<..$ the sequence of jump event with values in $\mathbb{R}^{2}_{+}$ and $\lambda(t)$ the intensity function at time $t$. We define the poisson random measure (PRM) associated to the sequence of jump event by

$$ M(\omega)(A)= \sum_{i\geq 0}\delta_{T_i(\omega)}(A),\quad A\in\mathcal{B}(\mathbb{R}^{2}) $$

I will skip the $\omega$ in the sequel for the PRM. Now we define the set

$$ A_t = \{ (x_1,x_2) : 0\leq x_1\leq t, 0\leq x_2\leq\lambda(x_1)\} $$

By precising that $\lambda(t) = \mu + \sum_{n : T_n <t}h(t-T_n)$ with $\mu>0$ and $h :\mathbb{R}_{+}\to\mathbb{R}_{+}$, we define the linear Hawkes process by

$$ Z_t = M(A_t) $$

Then, I have two concerns :

• The justification of such equality

$Z_t = \int_{[0,t]}\int_{0}^{\infty}1_{z\leq\lambda(s)} M(ds,dz)$

Indeed,I would like to start with the very definition of $Z_t$ that is with an integral over $\mathbb{R}^{2}$ since the PRM is a $2$ dimensional random measure :

$$ Z_t = \int_{[0,t]\times[0,\lambda(s)]}M(ds,dz) $$

But then I don’t know how to separate the integral in two since I cannot use Fubini in this context.

How can I justify the passage from the integral above to

$$ Z_t = \int_{[0,t]}\int_{0}^{\infty}1_{z\leq\lambda(s)} M(ds,dz) $$

?

This makes me doubt about my comprehension of this $Z_t$.

• The second concern is about the function $\lambda(t)$. Indeed $h$ is defined on $\mathbb{R}_{+}$ so the expression $h(t-T_n)$ does not make sense since $T_n(\omega)\in\mathbb{R}^{2}_{+}$ ?

This concerns me all the more as the following expression is often used

$$ \int_{(0,s)}h(s-u)dZ_u = \sum_{ k : T_k < s} h(s-T_k) $$

Which, for me until now, does not make sense given the definition of $T_n$.

If you have any answers that might help me, please don't hesitate! Thank you very much.

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$\mathbb{R}_{+}^{2} = \{ (x,y)\in\mathbb{R}^{2} : x\geq 0, y\geq 0\}$

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After some thoughts concerning the second point, it may be possible to consider as the initial point process $X_k$ with values in $\mathbb{R}^{2}_{+}$ (for example a $2$ dimensional uniform law) and then define

$$ T_k(\omega)= \pi_{1}(X_k(\omega))= x_1 $$

Where $X_k(\omega)=(x_1,x_2)$. In order to have

$$ \lambda(t) = \mu + \sum_{n : T_n <t}h(t-T_n) $$

that is well defined.

I don’t know if it is correct in that way but I think it avoids the confusion I highlighted above. Let me know what is your thoughts on this please.

Answer 1

As far as I can tell, applying Fubini shouldn't be an issue here: Note that, as you already described in your post, a Point Process is a random measure. Hence for any event $\omega$ the value $M(\omega)$ is a measure. Now the associated Hawkes Process reads $$M(A_t)(\omega)=\int 1_{A_t} dM(\omega)$$ Since the constant $1_{A_t}$ obviously is non-negative and the Point Process is defined on the product-$\sigma$-algebra $\mathbb B(\mathbb R^2_+)$, we can apply Fubini $\omega$-wise to obtain $$M(A_t)(\omega)=\int_0^\infty\int_0^\infty 1_{A_t}((x,y)) M(\omega)(dx,dy)=\int_0^t\int_0^{\lambda(s)}dM(\omega).$$ Hence the equality you described above holds for every $\omega$.

To the second point: As @Michh already mentioned, there seem to be different definitions of the Hawkes Process in the literature, either based on univariate Point Processes (as in mathworld.wolfram) or based on multivariate Point Processes, as you are doing in your post. The mathworld post also includes the expression you mentioned (see Eq. (2)), though their time points are 1-dimensional.

Regarding your setup: You suppose that $T_1<T_2<\dots<T_k$ are points in $\mathbb R^2_+$, but what does $T_i<T_{i+1}$ mean if the $T_i$ are multivariate? Component-wise inequality? You seem to be mixing up the two approaches here, but here is a connection between the "multivariate approach" and the "univariate approach":

Let $\lambda$ be an intensity function and let $N_t$ be an inhomogeneous Poisson Process with intensity $\lambda$, meaning $$\mathbb EN_t=\int_0^t\lambda(s)ds.$$ I will show how to construct this inhomogenous Poisson Process using a homogeneous Poisson Process. Let $\mathcal P$ be a homogeneous Poisson Point Process on $\mathbb R^2_+$, meaning the intensity measure of $\mathcal P$ is the Lebesgue measure. Then construct the set $A_t$ as you did: $$A_t:=\{(x_1,x_2)\in\mathbb R^2_+: x_1\leq t, x_2\leq \lambda(x_1)\}$$ and we see by Fubini that $$\mathbb E\big[\mathcal P(A_t)\big]=\int_0^t\int_0^{\lambda(s)}1 dz ds = \int_0^t\lambda(s)ds.$$ Thus $\mathcal P(A_t)$ has the same law as $N_t$. This argument can be repeated for general time intervals, showing that the laws of $(\mathcal P(A_t))_{t\geq 0}$ and $(N_t)_{t\geq 0}$ coincide.

Now the "one-dimensional appproach" is to construct an inhomogenous Poisson Process $N_t$ with conditional intensity $$\lambda(t):=\mu+\sum_{T_k<t}h(t-T_k)$$ where the time points $T_k$ are one-dimensional. The "multivariate approach" would be to consider a homogenous Poisson Point Process $\mathcal P$ as described above, then defining $M_t=\mathcal P(A_t)$. Note that the arguments given above still work for conditional intensities $\lambda$, since if $\mathcal F_t$ is the filtration up to time $t$, we have $$\frac{1}{h}\mathbb E\bigg[\mathcal P(A_{t+h})-\mathcal P(A_t)\big|\mathcal F_t\bigg]=\frac{1}{h}\int_t^{t+h}\int_0^{\lambda(s)}dzds=\frac{1}{h}\int_t^{t+h}\lambda(s)ds\to \lambda(t)$$ as $h\to 0$. Hence $\mathcal P(A_t)$ has the same conditional intensity as $N_t$. Since, under certain assumptions, the conditional intensity uniquely determines (the law of) a Poisson Process, we again get that the laws of $(\mathcal P(A_t))_{t\geq 0}$ and $(N_t)_{t\geq 0}$ coincide. For a reference, see these lecture notes, Proposition 2.2.

I hope my answer was somewhat helpful. Please correct me if you notice any mistakes!