# On the chaotic expansion for counting processes 333This research is
supported by a grant of the French National Research Agency (ANR),
“Investissements d’Avenir” (LabEx Ecodec/ANR-11-LABX-0047) and the Joint
Research Initiative ”Cyber Risk Insurance: actuarial modeling” with the
partnership of AXA Research Fund.

Caroline Hillairet111ENSAE Paris, CREST UMR 9194, 5 avenue Henry Le Chatelier
91120 Palaiseau, France. Email<EMAIL_ADDRESS>Anthony
Réveillac222INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135
avenue de Rangueil 31077 Toulouse Cedex 4 France. Email:
<EMAIL_ADDRESS>
###### Abstract

We introduce and study an alternative form of the chaotic expansion for
counting processes using the Poisson imbedding representation; we name this
alternative form pseudo-chaotic expansion. As an application, we prove that
the coefficients of this pseudo-chaotic expansion for any linear Hawkes
process are obtained in closed form, whereas those of the usual chaotic
expansion cannot be derived explicitly. Finally, we study further the
structure of linear Hawkes processes by constructing an example of a process
in a pseudo-chaotic form that satisfies the stochastic self-exciting intensity
equation which determines a Hawkes process (in particular its expectation
equals the one of a Hawkes process) but which fails to be a counting process.

Keywords: Counting processes; Poisson imbedding representation; Malliavin
calculus; Hawkes processes.
Mathematics Subject Classification (2020): 60G55; 60G57; 60H07.

##  1 Introduction

Counting processes constitute a mathematical framework for modeling specific
random events within a time series. The nature of the application and the
structural features of a given time series may call for sophisticated models
whose complexity goes beyond the standard Poisson process. For instance, the
modeling of neuron’s spikes in Neurosciences (see e.g. [3, 5]) or the
frequency of claims that may result in a cyber insurance contract (see [9] for
a short review on the literature) require counting processes with stochastic
intensity and whose frequency of the events depends on the past values of the
system. Hawkes processes initially introduced in [6] has become the paradigm
of such processes. The so-called linear Hawkes process is a counting process
$H$ (on a filtered probability space) with intensity process $\lambda$
satisfying :

$\lambda_{t}=\mu+\int_{(0,t)}\Phi(t-s)dH_{s},\quad t\geq 0,$ (1.1)

where the constant $\mu>0$ is the baseline intensity and
$\Phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is modeling the self-exciting feature
of the process. Naturally conditions on $\Phi$ are required for a well-posed
formulation. Well-posedness here is accurate as, from this formulation it
appears that the pair $(H,\lambda)$ solves a two-dimensional SDE driven by a
Poisson measure as we will make precise below. The so-called Poisson imbedding
provides a way to formulate this equation. Let
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $N$ a random
Poisson measure on $\mathbb{R}_{+}^{2}$ with intensity
$d\lambda(t,\theta):=dtd\theta$ the Lebesgue measure on $\mathbb{R}_{+}^{2}$.
If $\lambda$ denotes a non-negative predictable with respect to the natural
history of $N$ (whose definition will be recalled in Section 2) then the
process $H$ defined as

$H_{t}=\int_{(0,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\quad
t\geq 0,$ (1.2)

is a counting process with intensity $\lambda$, that is
$H-\int_{0}^{\cdot}\lambda_{s}ds$ is a martingale. The Poisson imbedding
refers to Representation (1.2) for counting processes. As mentioned, in case
of a linear Hawkes process for instance, the Poisson imbedding representation
(1.2) captures the equation feature of this process. Indeed combining (1.1)
and (1.2) implies that the linear Hawkes processes can be seen as a system of
weakly coupled SDEs with respect to $N$ as :

$\left\\{\begin{array}[]{l}H_{t}=\int_{(0,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta)\\\
\hskip 200.0003ptt\geq 0,\\\
\lambda_{t}=\mu+\int_{(0,t)}\Phi(t-s)\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta).\end{array}\right.$

Under mild condition on $\Phi$ the second equation (and so the system) can be
proved to be well-posed (see e.g. [3, 4, 9]).

This representation also opens the way to a new line of research. Indeed, with
this representation at hand, a counting process can then be seen as a
functional of the two-dimensional Poisson measure $N$ for which stochastic
analysis such as the Malliavin calculus is available. Recently, by combining
the Malliavin calculus with Stein’s method according to the Nourdin-Peccati
methodology, quantitative limit theorems for Hawkes functionals have been
derived in [14, 8, 10]. The specific Malliavin calculus developed in [8] for
linear Hawkes processes follows from a Mecke formula provided in [9] with
application to the risk analysis of a class of cyber insurance contracts.
Another main ingredient (not exploited so far in this context up to our
knowledge) when dealing with Gaussian and Poisson functionals is given by the
so-called chaotic expansion (also called Wiener-Itô expansion). Let $t\geq 0$
and $H_{t}$ as in (1.2). The chaotic expansion of $H_{t}$ with respect to $N$
writes down as :

$H_{t}=\mathbb{E}[H_{t}]+\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{[0,t]\times\mathbb{R}_{+}}\cdots\int_{[0,t]\times\mathbb{R}_{+}}f_{j}(x_{1},\ldots,x_{j})(N(dx_{1})-dx_{1})\cdots(N(dx_{j})-dx_{j}),$
(1.3)

where $x_{i}\in\mathbb{R}_{+}^{2}$ and $f_{j}$ is a symmetric function on
$(\mathbb{R}_{+}^{2})^{j}$ defined in terms of the $j$th Malliavin derivative
of $H_{t}$. This expression requires some details on its definition that will
be given in Section 2 below; but roughly speaking it allows one to expand the
random variable into iterated integrals with respect to the compensated
Poisson measure $N(dx_{j})-dx_{j}:=N(dt_{j},d\theta_{j})-dt_{j}d\theta_{j}$
(with $x_{j}:=(t_{j},\theta_{j})$). Such decomposition is proved to be useful
(for example in the context of Brownian SPDEs) provided that the coefficients
$f_{j}$ can be computed or can be characterized by an equation. In case of a
linear Hawkes process we will show in Section 4 that the coefficients can be
computed but are far from being explicit.

In this paper we prove as Theorem 3.15 that counting processes satisfy an
alternative representation of the chaotic expansion that we name pseudo-
chaotic representation. This pseudo chaotic expansion takes the form of :

$H_{t}=\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{[0,t]\times\mathbb{R}_{+}}\cdots\int_{[0,t]\times\mathbb{R}_{+}}c_{j}(x_{1},\ldots,x_{j})N(dx_{1})\cdots
N(dx_{j}),$ (1.4)

involving iterated integrals of the counting measure $N$ only (and not its
compensated version). Whereas any square integrable random variable $F$ admits
a chaotic expansion of the form (1.3) we characterize in Theorem 3.13 those
random variables for which a pseudo-chaotic expansion of the form (1.4) is
valid. As mentioned any variable $F=H_{t}$ with $H$ a counting process belongs
to this set.

In case where $H$ is a linear Hawkes process, in contradistinction to the
coefficients $f_{j}$ in the classical chaotic expansion (1.3) of $H_{t}$ at
some time $t$ which can not be computed explicitly, coefficients $c_{j}$ of
the pseudo-chaotic expansion (1.4) are explicit and given in Theorem 4.5 (see
Discussion 4.8). This provides then a closed-form expression to linear Hawkes
processes. Finally, we study further in Section 5 the structure of linear
Hawkes processes by constructing an example of a process in a pseudo-chaotic
form that satisfies the stochastic intensity equation (1.1) but which fails to
be a counting process (see Theorem 5.2 and Discussion 5.3).

The paper is organized as follows. Notations and the description of the
Poisson imbedding together with elements of Malliavin’s calculus are presented
in Section 2. The notion of pseudo-chaotic expansion is presented in Section
3. The application to linear Hawkes processes and their explicit
representation is given in Section 4. Finally, Section 5 is dedicated to the
construction of an example of a process in a pseudo-chaotic form that
satisfies the stochastic intensity equation but which fails to be a counting
process.

##  2 Preliminaries and notations

###  2.1 General conventions and notations

We set $\mathbb{N}^{*}:=\mathbb{N}\setminus\\{0\\}$ the set of positive
integers. We make use of the convention :

######  Convention 2.1.

For $a,b\in\mathbb{Z}$ with $a>b$, and for any map
$\rho:\mathbb{Z}\to\mathbb{R}$,

$\prod_{i=a}^{b}\rho(i):=1;\quad\sum_{i=a}^{b}\rho(i):=0.$

We set

$\mathbb{X}:=\mathbb{R}_{+}\times\mathbb{R}_{+}=\\{x=(t,\theta),\;t\in\mathbb{R}_{+},\;x\in\mathbb{R}_{+}\\};$
(2.1)

Throughout this paper we will make use of the notation $(t,\theta)$ to refer
to the first and second coordinate of an element in $\mathbb{X}$.

######  Notation 2.2.

Let $k\in\mathbb{N}^{*}$ and
$(x_{1},\ldots,x_{k})=((t_{1},\theta_{1}),\ldots,(t_{k},\theta_{k}))$ in
$\mathbb{X}^{k}$. We set $(x_{(1)},\ldots,x_{(k)})$ the ordered in the
$t$-component of $(x_{1},\ldots,x_{k})$ with $0\leq t_{(1)}\leq\cdots\leq
t_{(k)},$ and write $x_{(i)}:=(t_{(i)},\theta_{(i)})$.

We simply write $dx:=dt\,d\theta$ for the Lebesgue measure on $\mathbb{X}$. We
also set $\mathcal{B}(\mathbb{X})$ the set of Borelian of $\mathbb{X}$.

###  2.2 Poisson imbedding and elements of Malliavin calculus

Our approach lies on the so-called Poisson imbedding representation allowing
one to represent a counting process with respect to a baseline random Poisson
measure on $\mathbb{X}$. Most of the elements presented in this section are
taken from [13, 11].
We define $\Omega$ the space of configurations on $\mathbb{X}$ as

$\Omega:=\left\\{\omega=\sum_{i=1}^{n}\delta_{x_{i}},\;x_{i}:=(t_{i},\theta_{i})\in\mathbb{X},\;i=1,\ldots,n,\;0=t_{0}<t_{1}<\cdots<t_{n},\;\theta_{i}\in\mathbb{R}_{+},\;n\in\mathbb{N}\cup\\{+\infty\\}\right\\}.$

Each path of a counting process is represented as an element $\omega$ in
$\Omega$ which is a $\mathbb{N}$-valued $\sigma$-finite measure on
$\mathbb{X}=\mathbb{R}_{+}^{2}$. Let $\mathcal{F}$ be the $\sigma$-field
associated to the vague topology on $\Omega$. Let $\mathbb{P}$ the Poisson
measure on $\Omega$ under which the canonical process $N$ on $\Omega$ is a
Poisson process with intensity one that is :

$(N(\omega))([0,t]\times[0,b])(\omega):=\omega([0,t]\times[0,b]),\quad t\geq
0,\;b\in\mathbb{R}_{+},$

is an homogeneous Poisson process with intensity one ($N([0,t]\times[0,b])$ is
a Poisson random variable with intensity $bt$ for any $(t,b)\in\mathbb{X}$).
We set $\mathbb{F}^{N}:=({\mathcal{F}}_{t}^{N})_{t\geq 0}$ the natural history
of $N$, that is $\mathcal{F}_{t}^{N}:=\sigma\\{N(\mathcal{T}\times
B),\;\mathcal{T}\subset\mathcal{B}([0,t]),\;B\in\mathcal{B}(\mathbb{R}_{+})\\}$.
The expectation with respect to $\mathbb{P}$ is denoted by
$\mathbb{E}[\cdot]$. We also set
$\mathcal{F}_{\infty}^{N}:=\lim_{t\to+\infty}\mathcal{F}_{t}^{N}$.

####  2.2.1 Add-points operators and the Malliavin derivative

We introduce some elements of Malliavin calculus on Poisson processes.
For $n\in\mathbb{N}^{*}$, $f:\mathbb{X}^{n}\to\mathbb{R}$ is symmetric if for
any permutation $\sigma$ on $\\{1,\ldots,n\\}$, and for any
$(x_{1},\ldots,x_{n})\in\mathbb{X}^{n}$,
$f(x_{1},\ldots,x_{n})=f(x_{\sigma(1)},\ldots,x_{\sigma(n)})$. We set :

$L^{0}(\Omega):=\left\\{F:\Omega\to\mathbb{R},\;\mathcal{F}_{\infty}^{N}-\textrm{
measurable}\right\\},$ $L^{2}(\Omega):=\left\\{F\in
L^{0}(\Omega),\;\mathbb{E}[|F|^{2}]<+\infty\right\\}.$

Let for $j\in\mathbb{N}^{*}$

$L^{2}(\mathbb{X}^{j}):=\left\\{f:\mathbb{X}^{j}\to\mathbb{R},\;\int_{\mathbb{X}^{j}}|f(x_{1},\cdots,x_{j})|^{2}dx_{1}\cdots
dx_{j}<+\infty\right\\},$ (2.2)

and

$L^{2}_{s}(\mathbb{X}^{j}):=\left\\{f\textrm{ symmetric and }f\in
L^{2}(\mathbb{X}^{j})\right\\}$ (2.3)

the set of symmetric square integrable functions $f$ on $\mathbb{X}^{j}$.

For $h\in L^{2}(\mathbb{X})$ and $j\geq 1$ we set $h^{\otimes j}\in
L_{s}^{2}(\mathbb{X}^{j})$ defined as :

$h^{\otimes
j}(x_{1},\ldots,x_{j}):=\prod_{i=1}^{j}h(x_{i}),\quad(x_{1},\ldots,x_{j})\in\mathbb{X}^{j}.$
(2.4)

The main ingredient we will make use of are the add-points operators on the
Poisson space $\Omega$.

######  Definition 2.3.

$[$Add-points operators$]$

  * (i)

For $k$ in $\mathbb{N}^{*}$, and any subset of $\mathbb{X}$ of cardinal $k$
denoted $\\{x_{i},\;i\in\\{1,\ldots,k\\}\\}\subset\mathbb{X}$, we set the
measurable mapping :

$\displaystyle\varepsilon_{(x_{1},\ldots,x_{k})}^{+,k}:\Omega$
$\displaystyle\longrightarrow$ $\displaystyle\Omega$ $\displaystyle\omega$
$\displaystyle\longmapsto$ $\displaystyle\omega+\sum_{i=1}^{k}\delta_{x_{i}};$

with the convention that given a representation of $\omega$ as
$\omega=\sum_{i=1}^{n}\delta_{y_{i}}$ (for some $n\in\mathbb{N}^{*}$,
$y_{i}\in\mathbb{X}$), $\omega+\sum_{i=1}^{k}\delta_{x_{i}}$ is understood as
follows444Note that given fixed atoms $(x_{1},\ldots,x_{n})$, as $\mathbb{P}$
is the Poisson measure on $\Omega$, with $\mathbb{P}$-probability one, marks
$x_{i}$ do not belong to the representation of $\omega$. :

$\omega+\sum_{i=1}^{k}\delta_{x_{i}}:=\sum_{i=1}^{n}\delta_{y_{i}}+\sum_{i=1}^{k}\delta_{x_{i}}\textbf{1}_{\\{x_{i}\neq
y_{i}\\}}.$ (2.5)

  * (ii)

When $k=1$ we simply write
$\varepsilon_{x_{1}}^{+}:=\varepsilon_{x_{1}}^{+,1}$.

We now define the Malliavin derivative operator.

######  Definition 2.4.

For $F$ in $L^{2}(\Omega)$, $n\in\mathbb{N}^{*}$, and
$(x_{1},\ldots,x_{n})\in\mathbb{X}^{n}$, we set

$D_{(x_{1},\ldots,x_{n})}F:=F\circ\varepsilon_{(x_{1},\ldots,x_{n})}^{+,n}-F.$
(2.6)

For instance when $n=1$, we write $D_{x}F:=D_{x}^{1}F$ which is the difference
operator (also called add-one cost operator555see [11, p. 5]). Note that with
this definition, for any $\omega$ in $\Omega$, the mapping

$(x_{1},\ldots,x_{n})\mapsto D_{(x_{1},\ldots,x_{n})}^{n}F(\omega)$

is symmetric and belongs to $L^{2}_{s}(\mathbb{X}^{j})$ defined as (2.3).

We extend this definition to the iterated Malliavin derivatives.

######  Proposition 2.5.

(See e.g. [11, Relation (15)]) Let $F$ in $L^{2}(\Omega)$,
$n\in\mathbb{N}^{*}$, and $(x_{1},\ldots,x_{n})\in\mathbb{X}^{n}$. We set the
$n$th iterated Malliavin derivative operator $D^{n}$ as

$D^{n}F=D(D^{n-1}F),\quad n\geq 1;\quad D^{0}F:=F.$

It holds that

$D^{n}_{(x_{1},\ldots,x_{n})}F(\omega)=\sum_{J\subset\\{1,\cdots,n\\}}(-1)^{n-|J|}F\left(\omega+\sum_{j\in
J}\delta_{x_{j}}\right),\quad\textrm{ for a.e. }\omega\in\Omega,$

where the sum stands for all the subsets $J$ of $\\{1,\cdots,n\\}$ and $|J|$
denotes the cardinal of $J$.

######  Remark 2.6.

Note that $\omega+\sum_{j\in J}\delta_{x_{j}}$ is understood according to
(2.5).

####  2.2.2 Iterated integrals and the Chaotic expansion

The decompositions we are going to deal with take the form of iterated
stochastic integrals whose definition is made precise in this section.

######  Notation 2.7.

For $j\in\mathbb{N}^{*}$, $T>0$ and $M>0$ we set :

$\Delta_{j}:=\left\\{(x_{1},\cdots,x_{j})\in\mathbb{X}^{j},\;x_{i}\neq
x_{k},\;\forall i\neq k\in\\{1,\cdots,j\\}\right\\}.$

######  Definition 2.8.

Let $j\in\mathbb{N}^{*}$ and $f_{j}$ an element of
$L_{s}^{2}(\mathbb{X}^{j})$. We set $I_{j}(f_{j})$ the $j$th iterated integral
of $f_{j}$ against the compensated Poisson measure defined as :

$\displaystyle I_{j}(f_{j})$
$\displaystyle:=\int_{\Delta_{j}}f_{j}(x_{1},\ldots,x_{j})(N(dx_{1})-dx_{1})\cdots(N(dx_{j})-dx_{j})$
$\displaystyle=j!\int_{\mathbb{X}}\int_{[0,t_{j-1})\times\mathbb{R}_{+}}\cdots\int_{[0,t_{2})\times\mathbb{R}_{+}}f_{j}(x_{1},\ldots,x_{j})(N(dx_{1})-dx_{1})\cdots(N(dx_{j})-dx_{j})$
$\displaystyle=j!\int_{\mathbb{X}}\int_{[0,t_{j-1})\times\mathbb{R}_{+}}\cdots\int_{[0,t_{2})\times\mathbb{R}_{+}}f_{j}((t_{1},\theta_{1}),\ldots,(t_{j},\theta_{j}))(N(dt_{1},d\theta_{1})-dt_{1}d\theta_{1})\cdots(N(dt_{j},d\theta_{j})-dt_{j}d\theta_{j})$
(2.7)

where we recall the notation $x_{i}=(t_{i},\theta_{i})$ and
$dx_{i}=dt_{i}\,d\theta_{i}$. Recall that all the integrals are defined
pathwise.

These iterated integrals naturally appear in the chaotic expansion recalled
below.

######  Theorem 2.9 (See e.g. Theorem 2 in [11]).

Let $F$ in $L^{2}(\Omega)$. Then

$F=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(f_{j}^{F}),$

where the convergence of the series holds in $L^{2}(\omega,\mathbb{P})$ and
where the coefficients $f_{j}^{F}$ are the elements of
$L_{s}^{2}(\mathbb{X}^{j})$ (see (2.3)) given as

$\displaystyle f_{j}^{F}:\mathbb{X}^{j}$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{R}_{+}$ $\displaystyle(x_{1},\ldots,x_{j})$
$\displaystyle\longmapsto$
$\displaystyle\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right].$

In addition the decomposition is unique in the sense that : if there exist
elements $(g_{j})_{j\geq 1}$ with $g_{j}\in L_{s}^{2}(\mathbb{X}^{j})$ such
that

$F=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(g_{j})$

then $g_{j}=f_{j}^{F},\;dx-a.e.,\;\forall j\geq 1.$

This decomposition is similar to the Wiener-Itô decomposition on Gaussian
spaces. We conclude this section by recalling the link between the iterated
Malliavin derivative and the iterated integrals.

######  Theorem 2.10.

Let $j\in\mathbb{N}^{*}$, $g_{j}$ in $L^{2}_{s}(\mathbb{X}^{j})$ and $F$ in
$L^{2}(\Omega)$. Then :

$\mathbb{E}\left[\int_{\mathbb{X}^{j}}g_{j}(x_{1},\ldots,x_{j})\,D_{(x_{1},\ldots,x_{j})}^{j}Fdx_{1}\ldots
dx_{j}\right]=\mathbb{E}\left[FI_{j}(g_{j})\right].$

###### Proof.

This result is standard although we could not find a precise reference fitting
to our framework so we provide a sketch of the proof. To the Malliavin
derivative, one can associate its dual operator named the divergence operator
$\delta$ on a subset of the measurable elements
$u:\mathbb{X}\times\Omega\to\mathbb{R}$. More precisely, for any such $u$ in
$\textrm{Dom(}\delta\textrm{)}$ (the domain of the operator see for instance
[12, 13, 11]), $\delta(u)$ denotes the unique element in $L^{2}(\Omega)$ such
that :

$\mathbb{E}[F\delta(u)]=\int_{\mathbb{X}}\mathbb{E}[u(x)D_{x}F]dx,\quad\forall
F\in L^{2}(\Omega).$ (2.8)

By uniqueness, $\delta(u)$ coincides with the Itô stochastic integral in case
$u$ is a predictable process which itself is equal to $I_{1}(u)$ when $u$ is
deterministic. Similarly, the iterated divergence of order $j$ denoted
$\delta^{j}$ can be defined as the dual operator of the $j$th Malliavin
derivative $D^{j}$. In case of a deterministic element $g_{j}$,
$I_{j}(g_{j})=\delta^{j}(g_{j})$. To see this, note that for any $j\geq 2$ and
any $g\in L^{2}_{s}(\mathbb{X}^{j})$ it holds that

$I_{j}(g)=\delta(I_{j-1}(g(\cdot,\bullet))),\quad\cdot\in\mathbb{X},\;\bullet\in\mathbb{X}^{j-1},\;\textrm{
in }L^{2}(\Omega).$

Using then the Malliavin integration by parts formula (2.8) one gets the
result by induction. ∎

We conclude this section with the well-known relation between the Malliavin
derivatives and the iterated integrals that can be found for example in [11,
13].

######  Proposition 2.11.

  * (i)

Let $j\in\mathbb{N}^{*}$, $k\in\mathbb{N}^{*}$, with $k\leq j$ and $h\in
L^{2}(\mathbb{X})$. Then :

$D_{(x_{1},\cdots,x_{k})}^{k}I_{j}(h^{\otimes
j})=\frac{j!}{(j-k)!}I_{j-k}(h^{\otimes(j-k)})\prod_{i=1}^{k}h(x_{i}),\quad\forall(x_{1},\cdots,x_{k})\in\mathbb{X}^{k}.$

  * (ii)

In addition, for $k>j$,

$D_{(x_{1},\cdots,x_{k})}^{k}I_{j}(h^{\otimes
j})=0,\quad\forall(x_{1},\cdots,x_{k})\in\mathbb{X}^{k}.$

##  3 Notion of pseudo-chaotic expansion

We now present an alternative decomposition that we name pseudo-chaotic
expansion. Recall that according to Theorem 2.9, any $F$ in $L^{2}(\Omega)$
admits a chaotic expansion as :

$F=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(f_{j}^{F}),$

with
$f_{j}^{F}(x_{1},\cdots,x_{j})=\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]$.

For technical reasons we will also consider the same property but for a
baseline Poisson measure on a given bounded subset of $\mathbb{X}$.

######  Definition 3.1.

For $(T,M)\in\mathbb{X}$ we set :

  * (i)

$\Delta^{T,M}_{j}:=\left\\{(x_{1},\cdots,x_{j})\in([0,T]\times[0,M])^{j},\;x_{i}\neq
x_{k},\;\forall i\neq k\in\\{1,\cdots,j\\}\right\\},$

  * (ii)

$N^{T,M}$ the truncated Poisson measure $N^{T,M}$ defined as :

$N^{T,M}(A):=\int_{A}\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x)N(dx),\quad
A\in\mathcal{B}(\mathbb{X}).$

  * (iii)

$L^{2,T,M}(\Omega)$ the set of random variables $F$ in $L^{2}(\Omega)$ such
that there exists $(f_{j})_{j\geq 1}$ with $f_{j}\in
L_{s}^{2}(([0,T]\times[0,M])^{j})$ such that

$F=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(f_{j}),$

that is the set of random variables admitting the chaotic expansion with $N$
replaced by $N^{T,M}$.

######  Definition 3.2.

$[$Pseudo-chaotic expansion$]$

  1. 1.

A random variable $F$ in $L^{2}(\Omega)$ is said to have a pseudo-chaotic
expansion with respect to the counting process $N$ if there exists
$(g_{j})_{j\geq 1}$, $g_{j}\in L^{2}_{s}(\mathbb{X}^{j})$ for all
$j\in\mathbb{N}^{*}$ such that :

$F=\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{\mathbb{X}^{j}}g_{j}(x_{1},\ldots,x_{j})N(dx_{1})\cdots
N(dx_{j}),$ (3.1)

where the series converges in $L^{2}(\Omega)$.

  2. 2.

Fix $(T,M)\in\mathbb{X}$. A random variable $F$ in $L^{2,T,M}(\Omega)$ is said
to have a pseudo-chaotic expansion with respect to the counting process $N$ if
there exists $(g_{j})_{j\geq 1}$, $g_{j}\in L^{2}_{s}(([0,T]\times[0,M])^{j})$
for all $j\in\mathbb{N}^{*}$ such that :

$\displaystyle F$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{\mathbb{X}^{j}}g_{j}(x_{1},\ldots,x_{j})N^{T,M}(dx_{1})\cdots
N^{T,M}(dx_{j})$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{([0,T]\times[0,M])^{j}}g_{j}(x_{1},\ldots,x_{j})N(dx_{1})\cdots
N(dx_{j})$ (3.2)

where the series converges in $L^{2}(\Omega)$.

######  Remarks 3.3.

  * -

Recall that with the notations here above (and Notation 2.7), the symmetry of
functions $g_{j}$ entails that

$\displaystyle\int_{\mathbb{X}^{j}}g_{j}(x_{1},\ldots,x_{j})N(dx_{j})\cdots
N(dx_{1})$
$\displaystyle=\int_{\Delta_{j}}g_{j}(x_{1},\ldots,x_{j})N(dx_{1})\cdots
N(dx_{j})$
$\displaystyle=\int_{\Delta_{j}}g_{j}((t_{1},\theta_{1}),\ldots,(t_{j},\theta_{j}))N(dt_{1},d\theta_{1})\cdots
N(dt_{j},d\theta_{j})$
$\displaystyle=j!\int_{\mathbb{X}}\int_{[0,t_{j-1})\times\mathbb{R}_{+}}\cdots\int_{[0,t_{2})\times\mathbb{R}_{+}}g_{j}((t_{1},\theta_{1}),\ldots,(t_{j},\theta_{j}))N(dt_{1},d\theta_{1})\cdots
N(dt_{j},d\theta_{j}).$

  * -

Note that in each term
$\int_{\mathbb{X}^{j}}g_{j}(x_{1},\ldots,x_{j})N(dx_{1})\cdots N(dx_{j})$ in
(3.1), the multiple integration coincides with the one with respect to the so-
called factorial measures presented in [11, Appendix].

######  Definition 3.4.

We set

$\mathcal{P}:=\\{F\in L^{2}(\Omega),\textrm{ which admits a pseudo-chaotic
expansion with respect to }N\\},$

and for $(T,M)\in\mathbb{X}$,

$\mathcal{P}^{T,M}:=\\{F\in L^{2}(\Omega),\textrm{ which admits a pseudo-
chaotic expansion with respect to }N^{T,M}\\}.$

Before studying and characterizing those random variables which admit a
pseudo-chaotic expansion we need some preliminary results collected in the
section below.

###  3.1 Some preliminary results

######  Definition-Proposition 3.5.

For $(T,M)\in\mathbb{X}$.

  * (i)

Let the random variable $L^{T,M}$ on $\Omega$,

$L^{T,M}(\omega):=\exp\left(MT\right)\textbf{1}_{\\{N([0,T]\times[0,M])(\omega)=0\\}}=\exp\left(MT\right)\textbf{1}_{\\{\omega([0,T]\times[0,M])=0\\}},\quad\omega\in\Omega.$
(3.3)

It holds that

$L^{T,M}=1+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}),$

and

$D_{(x_{1},\cdots,x_{j})}^{j}L^{T,M}=L^{T,M}\prod_{i=1}^{j}(-\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x_{i})),\quad\forall(x_{1},\cdots,x_{j})\in\mathbb{X}^{j}.$
(3.4)

  * (ii)

$\mathbb{Q}^{T,M}$ defined on $(\Omega,\mathcal{F}_{T}^{N})$ as
$\frac{d\mathbb{Q}^{T,M}}{d\mathbb{P}}=:L^{T,M}$ is a probability measure.

As the support of $\mathbb{Q}^{T,M}$ is contained in
$\\{N([0,T]\times[0,M])=0\\}$ we name $\mathbb{Q}^{T,M}$ the vanishing Poisson
measure as it brings the intensity of $N$ to $0$ on the rectangle
$[0,T]\times[0,M]$ and

$\mathbb{E}^{\mathbb{Q}^{T,M}}[N^{T,M}(A)]=0,\quad\forall
A\in\mathcal{B}(\mathbb{X}).$

###### Proof.

Set

$L=1+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}).$

Following [13, Proposition 6.3.1], this expression is the chaotic expansion of
the stochastic exponential at time $T$ of the deterministic function
$x\mapsto-\textbf{1}_{\\{(0,T)\times[0,M]\\}}(x)$ against the compensated
Poisson measure, that is

$\displaystyle L$
$\displaystyle=\exp\left(\int_{\mathbb{X}}-\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x)\tilde{N}(dx)\right)\prod_{x,\;N(\\{x\\})=1}\left[(1-\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x))\exp\left(\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x)\right)\right]$
$\displaystyle=\exp\left(MT\right)\textbf{1}_{\\{N([0,T]\times[0,M])=0\\}}$
$\displaystyle=L^{T,M}.$

In addition, by definition of the Malliavin derivative $D$ and the exponential
structure of $L$ we have that

$D_{x_{1}}L=-L\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x_{1}).$

Relation (3.4) then results from the fact that
$-\textbf{1}_{\\{[0,T]\times[0,M]\\}}$ is deterministic and that
$D^{j}=D^{j-1}D$.
Finally, as $\mathbb{E}[L^{T,M}=1]$, the measure $\mathbb{Q}^{T,M}$ in (ii) is
well defined and is a probability measure (not equivalent to $\mathbb{P}$). ∎

######  Remark 3.6.

Our analysis will be based on the intervention of the quantity $L^{T,M}$ or of
the Poisson vanishing measure $\mathbb{Q}^{T,M}$ which is properly defined
only on bounded subsets of $\mathbb{X}$; which explains why we derive our
results for the truncated Poisson measures $N^{T,M}$ and not for $N$.

######  Proposition 3.7.

Let $(T,M)\in\mathbb{X}$, $j\in\mathbb{N}^{*}$, $f\in
L_{s}^{2}(([0,T]\times[0,M])^{j})$. Let $F$ of the form (2) (that is
$F\in\mathcal{P}^{T,M}$). Then

$\mathbb{E}^{\mathbb{Q}^{T,M}}\left[F\right]=\mathbb{E}\left[L^{T,M}F\right]=0.$

###### Proof.

The result is an immediate consequence of the definition of $\mathbb{Q}^{T,M}$
which is supported on the set $N([0,T]\times[0,M])=0$ and of the form of $F$
as a sum (2) involving only integrals against $N$ on $[0,T]\times[0,M]$. ∎

Our first main result which gives a motivation to the definition of the notion
of the pseudo-chaotic expansion lies in the theorem below.

######  Theorem 3.8.

Let $T,M>0$ and $F$ in $L^{2,T,M}(\Omega)$ (see (ii) of Definition 3.1). Then

$\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{\mathbb{X}^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}=\mathbb{E}\left[F(L^{T,M}-1)\right].$

###### Proof.

Using the Malliavin integration by parts it holds that :

$\displaystyle\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{\mathbb{X}^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}$
$\displaystyle\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{([0,T\times[0,M])^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\mathbb{E}\left[\int_{([0,T\times[0,M])^{j}}D_{(x_{1},\cdots,x_{j})}^{j}F\;dx_{1}\cdots
dx_{j}\right]$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\mathbb{E}\left[\int_{\mathbb{X}^{j}}\prod_{i=1}^{j}\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x_{i})D_{(x_{1},\cdots,x_{j})}^{j}F\;dx_{1}\cdots
dx_{j}\right]$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{1}{j!}\mathbb{E}\left[\int_{\mathbb{X}^{j}}\prod_{i=1}^{j}(-\textbf{1}_{\\{[0,T]\times[0,M]\\}}(x_{i}))D_{(x_{1},\cdots,x_{j})}^{j}Fdx_{1}\cdots
dx_{j}\right]$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{1}{j!}\mathbb{E}\left[\int_{\mathbb{X}^{j}}(-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j})(x_{1},\cdots,x_{j})D_{(x_{1},\cdots,x_{j})}^{j}Fdx_{1}\cdots
dx_{j}\right]$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{1}{j!}\mathbb{E}\left[F\;I_{j}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}\right)\right],\quad\textrm{ by Theorem \ref{theorem:generalizedIPP}}$
$\displaystyle=\mathbb{E}\left[F(L^{T,M}-1)\right].$

∎

The previous result will find interest for instance when $F=H_{T}$ for $H$ a
counting process and $T>0$. We make precise the definition of such processes.

######  Definition 3.9.

$[$Counting process with bounded intensity by Poisson imbedding$]$
Let $\lambda$ be a $\mathcal{F}^{N}$-predictable process such that

$\exists M>0,\textrm{ such that }\;\lambda_{t}\leq M,\quad\forall t\geq
0,\;\mathbb{P}-a.s..$

The process $H$ defined below is a counting process with intensity $\lambda$ :

$H_{t}=\int_{(0,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta)=\int_{(0,t]\times[0,M]}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\quad
t\geq 0.$ (3.5)

Using the chaotic expansion we have for any $T>0$ that :

$H_{T}=\mathbb{E}[H_{T}]+\sum_{j\geq
1}\frac{1}{j!}I_{j}(f_{j}^{H_{T}}),\quad\textrm{ with }\quad
f_{j}^{H_{T}}(x_{1},\cdots,x_{j})=\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}H_{T}\right],\;x_{i}\in[0,T]\times[0,M].$
(3.6)

We now apply Theorem 3.8 to a counting process.

######  Corollary 3.10.

Let $T>0$, $H$ a counting process with bounded intensity $\lambda$ by some
$M>0$ so that $H_{T}$ is given by (3.5). Then

$\displaystyle\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{\mathbb{X}^{j}}f_{j}^{H_{T}}(x_{1},\cdots,x_{j})dx_{1}\cdots
dx_{j}$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{\mathbb{X}^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}H_{T}\right]dx_{1}\cdots
dx_{j}$ $\displaystyle=-\mathbb{E}[H_{T}].$

###### Proof.

By Theorem 3.8,

$\displaystyle\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{([0,T]\times[0,M])^{j}}f_{j}^{H_{T}}(x_{1},\cdots,x_{j})dx_{1}\cdots
dx_{j}$
$\displaystyle=\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{([0,T]\times[0,M])^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}H_{T}\right]dx_{1}\cdots
dx_{j}$ $\displaystyle=\mathbb{E}\left[H_{T}(L^{T,M}-1)\right].$

Proposition 3.7 entails then that

$\mathbb{E}\left[H_{T}L^{T,M}\right]=\exp(MT)\mathbb{E}\left[H_{T}\textbf{1}_{\\{N([0,T]\times[0,M])=0\\}}\right]=0$

which concludes the proof. ∎

######  Remark 3.11.

The previous result is at the core of our analysis. This means that for a
counting process with bounded intensity, the only term with only Lebesgue
integrals ($dx$) in Expansion (3.6) vanishes with $\mathbb{E}[H_{T}]$. In
other words, all the terms in Expansion (3.6) involves at least on integral
against $N$.

We conclude this section with a generalized version of Theorem 3.8.

######  Lemma 3.12.

Fix $(T,M)\in\mathbb{X}$ and $F$ in $L^{2,T,M}(\Omega)$. For any
$k\in\mathbb{N}^{*}$ and any $(x_{1},\cdots,x_{k})\in([0,T]\times[0,M])^{k}$,
it holds that :

$\displaystyle\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})}^{j}F\right]dx_{j}\cdots
dx_{k+1}$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}FL^{T,M}\right].$

###### Proof.

The property $D^{j}=D^{j-k}D^{k}$ (for the first equality) and Theorem 2.10
applied to $D_{(x_{1},\ldots,x_{k})}^{k}F$ (for the second equality) imply

$\displaystyle\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})}^{j}F\right]dx_{j}\cdots
dx_{k+1}$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}\mathbb{E}\left[D_{(x_{k+1},\ldots,x_{j})}^{j-k}D_{(x_{1},\ldots,x_{k})}^{k}F\right]dx_{j}.$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{1}{j!}\frac{j!}{(j-k)!}\mathbb{E}\left[I_{j-k}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes(j-k)}\right)D_{(x_{1},\ldots,x_{k})}^{k}F\right]$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\mathbb{E}\left[\sum_{j=k+1}^{+\infty}\frac{1}{j!}\frac{j!}{(j-k)!}I_{j-k}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes(j-k)}\right)D_{(x_{1},\ldots,x_{k})}^{k}F\right].$

Then (i) of Proposition 2.11 entails that

$\displaystyle\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})}^{j}F\right]dx_{j}\cdots
dx_{k+1}$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\mathbb{E}\left[\sum_{j=k+1}^{+\infty}\frac{1}{j!}(-1)^{k}\left(D_{(x_{1},\ldots,x_{k})}^{k}I_{j}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}\right)\right)D_{(x_{1},\ldots,x_{k})}^{k}F\right]$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+(-1)^{k}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\left(D_{(x_{1},\ldots,x_{k})}^{k}\sum_{j=k+1}^{+\infty}\frac{1}{j!}I_{j}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}\right)\right)\right]$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+(-1)^{k}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\left(D_{(x_{1},\ldots,x_{k})}^{k}\left(L^{T,M}-1-\sum_{j=1}^{k}\frac{1}{j!}I_{j}\left((-\textbf{1}_{\\{[0,T]\times[0,M]\\}})^{\otimes
j}\right)\right)\right)\right],$

where we have used Proposition-Definition 3.5. Thus (i) and (ii) of
Proposition 2.11 give

$\displaystyle\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})}^{j}F\right]dx_{j}\cdots
dx_{k+1}$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+(-1)^{k}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\left(D_{(x_{1},\ldots,x_{k})}^{k}L^{T,M}+(-1)^{k+1}\right)\right]$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+(-1)^{k}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\left((-1)^{k}L^{T,M}+(-1)^{k+1}\right)\right]$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\left(L^{T,M}-1\right)\right]$
$\displaystyle=\mathbb{E}\left[L^{T,M}D_{(x_{1},\ldots,x_{k})}^{k}F\right].$

∎

###  3.2 Characterization of $\mathcal{P}^{T,M}$

Throughout this section we fix $(T,M)$ in $\mathbb{X}$. Corollary 3.10
suggests that random variables of the form $H_{T}$ with $H$ a counting process
satisfy the pseudo-chaotic expansion. We make precise this point and
characterize the set $\mathcal{P}^{T,M}$.

######  Theorem 3.13.

An element $F$in $L^{2}(\Omega)$ admits a pseudo-chaotic expansion with
respect to the counting process $N^{T,M}$ (that is $F\in\mathcal{P}^{T,M}$) if
and only if

$\mathbb{E}[F]=\sum_{j=1}^{+\infty}\frac{(-1)^{j+1}}{j!}\int_{([0,T]\times[0,M])^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}.$ (3.7)

In that case the pseudo-chaotic expansion of $F$ is given by

$F=\sum_{k=1}^{+\infty}\int_{([0,T]\times[0,M])^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$ (3.8)

with

$c_{k}(x_{1},\ldots,x_{k}):=\mathbb{E}\left[L^{T,M}\,D_{(x_{1},\ldots,x_{k})}^{k}F\right]=\mathbb{E}^{\mathbb{Q}^{T,M}}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right],\quad\forall(x_{1},\ldots,x_{k})\in([0,T]\times[0,M])^{k}.$
(3.9)

###### Proof.

Let $F$ in $\mathcal{P}^{T,M}$. Then, according to Theorem 3.8 and Proposition
3.7

$\displaystyle\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\int_{([0,T]\times[0,M])^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}$ $\displaystyle=\mathbb{E}\left[F(L^{T,M}-1)\right]$
$\displaystyle=\mathbb{E}^{\mathbb{Q}^{T,M}}[F]-\mathbb{E}[F]$
$\displaystyle=-\mathbb{E}[F].$

So

$\sum_{j=1}^{+\infty}\frac{(-1)^{j+1}}{j!}\int_{([0,T]\times[0,M])^{j}}\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]dx_{1}\cdots
dx_{j}=\mathbb{E}\left[F\right].$

Conversely assume $F\in L^{2}(\Omega)$ is such that Relation (3.7) is true.
The chaotic expansion (see Theorem 2.9) allows one to write

$F=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(\mathbb{E}[D^{j}F]).$

The definition of the iterated integrals $I_{j}$ together with Relation (3.7)
implies then that

$\displaystyle F$
$\displaystyle=\mathbb{E}[F]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(f_{j}^{F}),\quad\textrm{
with }\quad
f_{j}^{F}(x_{1},\cdots,x_{j})=\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}F\right]$
$\displaystyle=\sum_{k=1}^{+\infty}\frac{1}{k!}\int_{([0,T]\times[0,M])^{k}}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$

with

$\displaystyle c_{k}(x_{1},\ldots,x_{k})$
$\displaystyle:=f_{k}^{F}(x_{1},\ldots,x_{k})$
$\displaystyle+{k!}\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{{k!}(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}f_{j}^{F}(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})dx_{k+1}\cdots
dx_{j}.$

Here the number $\frac{j!}{{k!}(j-k)!}$ of $k$-combinations among $j$ choices
denotes the number of times the integral
$\int_{([0,T]\times[0,M])^{j-k}}f_{j}^{F}(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})dx_{k+1}\cdots
dx_{j}$ of the symmetric function $f_{j}^{F}$ appears in the expansion of
$I_{j}(f_{j}^{F})$. Note also the choice of normalisation by factoring $c_{k}$
with $\frac{1}{k!}$ which explains the $k!$ factor in front of the sum. We now
compute each of these terms. Using the definition of the $f_{j}^{F}$ functions
and Lemma 3.12 we get

$\displaystyle c_{k}(x_{1},\ldots,x_{k})$
$\displaystyle=f_{k}^{F}(x_{1},\ldots,x_{k})+\sum_{j\geq
k+1}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{\mathbb{X}^{j-k}}f_{j}^{F}(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})dx_{k+1}\cdots
dx_{j}$
$\displaystyle=\mathbb{E}\left[D_{(x_{1},\ldots,x_{k})}^{k}F\right]+\sum_{j\geq
k+1}\frac{(-1)^{j-k}}{j!}\frac{j!}{(j-k)!}\int_{\mathbb{X}^{j-k}}\mathbb{E}\left[D_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{j})}^{j}F\right]dx_{k+1}\cdots
dx_{j}$
$\displaystyle=\mathbb{E}\left[L^{T,M}D_{(x_{1},\ldots,x_{k})}^{k}F\right].$

∎

######  Remark 3.14.

Note that the uniqueness of the coefficients in the chaotic expansion
transfers to the uniqueness of the pseudo-chaotic expansion when it exists and
is given by the coefficients $c_{k}$ in (3.9).

We now apply this result to counting processes with bounded intensity
processes.

######  Theorem 3.15.

Let $T>0$, $H$ a counting process with bounded intensity $\lambda$ by $M>0$ so
that $H_{T}$ is given by (3.5). Then $H_{T}$ admits a pseudo-chaotic expansion
with respect to $N^{T,M}$ with

$H_{T}=\sum_{k=1}^{+\infty}\int_{([0,T]\times[0,M])^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$ (3.10)
$c_{k}(x_{1},\ldots,x_{k}):=\mathbb{E}\left[L^{T,M}D_{(x_{(1)},\ldots,x_{(k-1)})}^{k-1}\textbf{1}_{\\{\theta_{(k)}\leq\lambda_{(t_{k})}\\}}\right],\quad\forall(x_{1},\ldots,x_{k})\in([0,T]\times[0,M])^{k}$
(3.11)

where according to Notation 2.2, $0\leq t_{(1)}\leq\cdots\leq t_{(k)}\leq T$
are the ordered elements $(t_{1},\ldots,t_{k})$ and
$x_{(i)}:=(t_{(i)},\theta_{(i)})$.

###### Proof.

Theorem 3.10 and Theorem 3.13 give that $H_{T}$ admits a pseudo-chaotic
expansion and

$H_{T}=\sum_{k=1}^{+\infty}\int_{([0,T]\times[0,M])^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$

with
$c_{k}(x_{1},\ldots,x_{k}):=\mathbb{E}\left[L^{T,M}D_{(x_{1},\ldots,x_{k})}^{k}H_{T}\right],\quad\forall(x_{1},\ldots,x_{k})\in([0,T]\times[0,M])^{k}.$
Let $k\geq 1$ and $(x_{1},\ldots,x_{k})$ in $([0,T]\times[0,M])^{k}$. As
$(x_{1},\cdots,x_{k})\mapsto D_{(x_{1},\ldots,x_{k})}^{k}H_{T}$ is symmetric,

$D_{(x_{1},\ldots,x_{k})}^{k}H_{T}=D_{(x_{(1)},\ldots,x_{(k)})}^{k}H_{T}=D_{x_{1}}\cdots
D_{x_{k}}H_{T}.$

Using the definition of $D$ and using the fact that

$H_{T}=\int_{(0,T]\times[0,M]}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta)$

one gets that

$D_{(x_{(1)},\ldots,x_{(k)})}^{k}H_{T}=D_{(x_{(1)},\ldots,x_{(k-1)})}^{k-1}\textbf{1}_{\\{\theta_{(k)}\leq\lambda_{(t_{k})}\\}}+\int_{[0,T]\times\mathbb{R}_{+}}D_{(x_{(1)},\ldots,x_{(k)})}^{k}\textbf{1}_{\\{\theta\leq\lambda_{t}\\}}N(dt,d\theta).$
(3.12)

As $L^{T,M}$ annihilates the Poisson process on $[0,T]\times[0,M]$ it holds
that

$\mathbb{E}\left[L^{T,M}D_{(x_{1},\ldots,x_{k})}^{k}H_{T}\right]=\mathbb{E}\left[L^{T,M}D_{(x_{(1)},\ldots,x_{(k-1)})}^{k-1}\textbf{1}_{\\{\theta_{(k)}\leq\lambda_{(t_{k})}\\}}\right].$

∎

##  4 Application to linear Hawkes processes

Throughout this section $\Phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ denotes a map
in $L^{1}(\mathbb{R}_{+};dt)$.

###  4.1 Generalities on linear Hawkes processes

######  Assumption 4.1.

The mapping $\Phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ belongs to
$L^{1}(\mathbb{R}_{+};dt)$ with

$\|\Phi\|_{1}:=\int_{\mathbb{R}_{+}}\Phi(t)dt<1.$

For $f,g$ in $L^{1}(\mathbb{R}_{+};dt)$ we define the convolution of $f$ and
$g$ by

$(f\ast g)(t):=\int_{0}^{t}f(t-u)g(u)du,\quad t\geq 0.$

######  Proposition 4.2 (See e.g. [1]).

Assume $\Phi$ enjoys Assumption 4.1. Let

$\Phi_{1}:=\Phi,\quad\Phi_{n}(t):=\int_{0}^{t}\Phi(t-s)\Phi_{n-1}(s)ds,\quad
t\in\mathbb{R}_{+},\;n\in\mathbb{N}^{*}.$ (4.1)

For every $n\geq 1$, $\|\Phi_{n}\|_{1}=\|\Phi\|_{1}^{n}$ and the mapping
$\Psi:=\sum_{n=1}^{+\infty}\Phi_{n}$ is well-defined as a limit in
$L_{1}(\mathbb{R}_{+};dt)$ and
$\|\Psi\|_{1}=\frac{\|\Phi\|_{1}}{1-\|\Phi\|_{1}}$.

######  Definition 4.3 (Linear Hawkes process, [6]).

Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F}:=(\mathcal{F}_{t})_{t\geq 0})$
be a filtered probability space, $\mu>0$ and
$\Phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfying Assumption 4.1. A linear
Hawkes process $H:=(H_{t})_{t\geq 0}$ with parameters $\mu$ and $\Phi$ is a
counting process such that

  * (i)

$H_{0}=0,\quad\mathbb{P}-a.s.$,

  * (ii)

its ($\mathbb{F}$-predictable) intensity process is given by

$\lambda_{t}:=\mu+\int_{(0,t)}\Phi(t-s)dH_{s},\quad t\geq 0,$

that is for any $0\leq s\leq t$ and $A\in\mathcal{F}_{s}$,

$\mathbb{E}\left[\textbf{1}_{A}(H_{t}-H_{s})\right]=\mathbb{E}\left[\int_{(s,t]}\textbf{1}_{A}\lambda_{r}dr\right].$

###  4.2 Pseudo-chaotic expansion of linear Hawkes processes and explicit
representation

We aim at providing the coefficients in the pseudo-chaotic expansion of a
linear Hawkes process. We start with some general facts regarding a linear
Hawkes process.

######  Proposition 4.4.

Let $\Phi$ as in Assumption 4.1 and $\mu>0$ and $(H,\lambda)$ the Hawkes
process defined as the unique solution to the SDE

$\left\\{\begin{array}[]{l}H_{t}=\int_{(0,t]\times\mathbb{R}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\\\
\\\ \lambda_{t}=\mu+\int_{(0,t)\times\mathbb{R}_{+}}\Phi(t-s)dH_{s},\quad
t\geq 0\end{array}\right.$

Let $n\in\mathbb{N}^{*}$ and
$\\{y_{1},\ldots,y_{n}\\}=\\{(s_{1},\theta_{1}),\ldots,(s_{n},\theta_{n})\\}\subset\mathbb{X}$
with $0\leq s_{1}\leq\cdots\leq s_{n}\leq t$.
We set
$(a_{1}^{\\{y_{1},\ldots,y_{n}\\}},\cdots,a_{n}^{\\{y_{1},\ldots,y_{n}\\}})$
the solution to the system

$\left\\{\begin{array}[]{l}a_{1}^{\\{y_{1},\ldots,y_{n}\\}}=\mu+\textbf{1}_{\\{\theta_{1}\leq\mu\\}},\\\
\\\
a_{j}^{\\{y_{1},\ldots,y_{n}\\}}=\mu+\displaystyle{\sum_{i=1}^{j-1}\Phi(s_{j}-s_{i})\textbf{1}_{\\{\theta_{i}\leq
a_{i}^{\\{y_{1},\ldots,y_{n}\\}}\\}}},\quad
j\in\\{2,\ldots,k\\}.\end{array}\right.$ (4.2)

which is the triangular system

$\left\\{\begin{array}[]{l}a_{1}^{\\{y_{1},\ldots,y_{n}\\}}=\mu+\textbf{1}_{\\{\theta_{1}\leq\mu\\}},\\\
\\\
a_{2}^{\\{y_{1},\ldots,y_{n}\\}}=\mu+\displaystyle{\Phi(s_{2}-s_{1})\textbf{1}_{\\{\theta_{2}\leq
a_{1}^{\\{y_{1},\ldots,y_{n}\\}}\\}}},\\\ \\\ \hskip 50.00008pt\vdots\\\ \\\
a_{n}^{\\{y_{1},\ldots,y_{n}\\}}=\mu+\displaystyle{\sum_{i=1}^{n-1}\Phi(s_{n}-s_{i})\textbf{1}_{\\{\theta_{i}\leq
a_{i}^{\\{y_{1},\ldots,y_{n}\\}}\\}}}.\end{array}\right.$ (4.3)

Let
$\varpi_{\\{y_{1},\ldots,y_{n}\\}}:=\sum_{i=1}^{n}\delta_{y_{i}}\in\Omega.$
Then the values of the deterministic path
$\lambda(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$ (resulting from the evaluation of
$\lambda$ at the specific $\omega=\varpi_{\\{y_{1},\ldots,y_{n}\\}}$) at times
$s_{1},\ldots,s_{n}$ is given by

$(\lambda_{s_{1}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}}),\ldots,\lambda_{s_{n}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}}))=(a_{1}^{\\{y_{1},\ldots,y_{n}\\}},\ldots,a_{n}^{\\{y_{1},\ldots,y_{n}\\}}).$

In addition

$\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu+\sum_{i=1}^{n-1}\Phi(t-s_{i})\textbf{1}_{\\{\theta_{i}\leq
a_{i}^{\\{y_{1},\ldots,y_{n}\\}}\\}}\textbf{1}_{\\{s_{i}<t\\}},\quad\forall
t\geq s_{n}.$ (4.4)

###### Proof.

Let $t\geq 0$. By definition of $\lambda$,

$\displaystyle\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$
$\displaystyle:=\mu+\left(\int_{(0,t)}\Phi(t-u)dH_{u}\right)(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$
$\displaystyle=\mu+\left(\int_{(0,t)}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}\\}}N(du,d\theta)\right)(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$
$\displaystyle=\mu+\int_{(0,t)}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}(N(du,d\theta)(\varpi_{\\{y_{1},\ldots,y_{n}\\}}))$
$\displaystyle=\mu+\int_{(0,t)}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})(du,d\theta)$
$\displaystyle=\mu+\int_{(0,s_{1})}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}\textbf{1}_{\\{u<t\\}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})(du,d\theta)$
$\displaystyle+\sum_{i=1}^{n-1}\int_{[s_{i},s_{i+1})}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}\textbf{1}_{\\{u<t\\}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})(du,d\theta)$
$\displaystyle=\mu+\sum_{i=1}^{n-1}\int_{[s_{i},s_{i+1})}\Phi(t-u)\textbf{1}_{\\{\theta\leq\lambda_{u}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}\textbf{1}_{\\{u<t\\}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})(du,d\theta)$
$\displaystyle=\mu+\sum_{i=1}^{n-1}\Phi(t-s_{i})\textbf{1}_{\\{\theta_{i}\leq\lambda_{s_{i}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}\textbf{1}_{\\{s_{i}<t\\}}.$

In addition, by definition, of $\lambda$, for any $i$,
$\lambda_{s_{i}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\lambda_{s_{i}}(\varpi_{(y_{1},\ldots,y_{i-1})})$.
Hence, the evaluation of $\lambda$ at the specific path
$\varpi_{\\{y_{1},\ldots,y_{n}\\}}$ is the deterministic path completely
determined by its value at the dates $s_{1},\ldots,s_{n}$. Indeed,

$\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu,\quad\forall
t\in[0,s_{1}],$

in particular $a_{1}:=\lambda_{s_{1}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu$.
From this we deduce that for $t\in(s_{1},s_{2}]$,

$\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu+\Phi(t-s_{1})\textbf{1}_{\\{\theta_{1}\leq\lambda_{s_{1}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}=\mu+\Phi(t-s_{1})\textbf{1}_{\\{\theta_{1}\leq\mu\\}}.$

In particular
$a_{2}:=\lambda_{s_{2}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu+\Phi(s_{2}-s_{1})\textbf{1}_{\\{\theta_{1}\leq\mu\\}}=\mu+\Phi(s_{2}-s_{1})\textbf{1}_{\\{\theta_{1}\leq
a_{2}\\}}$. By induction we get that for $t\in(s_{j},s_{j+1}]$
($j\in\\{1,\cdots,n-1\\}$),

$\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})=\mu+\sum_{i=1}^{j}\Phi(t-s_{i})\textbf{1}_{\\{\theta_{i}\leq
a_{i}\\}}\textbf{1}_{\\{s_{i}<t\\}},$

with $a_{i}:=\lambda_{s_{i}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$. In other
words, $(a_{1},\ldots,a_{n})$ solves the triangular system of the statement. ∎

######  Theorem 4.5.

$[$Pseudo-chaotic expansion for linear Hawkes processes$]$
Let $\Phi$ as in Assumption 4.1 and $\mu>0$. Assume in addition that666with
classical notations $\|\Phi\|_{\infty}:=\sup_{t\geq 0}\Phi(t)$
$\|\Phi\|_{\infty}<+\infty$. Let $(H,\lambda)$ be the unique solution of

$\left\\{\begin{array}[]{l}H_{t}=\int_{(0,t]\times\mathbb{R}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\\\
\\\ \lambda_{t}=\mu+\int_{(0,t)\times\mathbb{R}_{+}}\Phi(t-s)dH_{s},\quad
t\geq 0\end{array}\right.$ (4.5)

Then $H$ is a linear Hawkes process with intensity $\lambda$ in the sense of
Definition 4.3. For any $T>0$, $H_{T}$ admits the pseudo-chaotic expansion
below :

$H_{T}=\sum_{k=1}^{+\infty}\int_{\mathbb{X}^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$ (4.6)
$\left\\{\begin{array}[]{l}c_{1}(x_{1})=\textbf{1}_{\\{\theta_{1}\leq\mu\\}},\\\
\\\
c_{k}(x_{1},\ldots,x_{k})=\displaystyle{(-1)^{k-1}\textbf{1}_{\\{\theta_{k}\leq\mu\\}}+\sum_{n=1}^{k-1}\sum_{\\{y_{1},\ldots,y_{n}\\}\subset\\{x_{(1)},\ldots,x_{(k-1)}\\}}(-1)^{k-1-n}\textbf{1}_{\\{\theta_{k}\leq\lambda_{x_{k}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})\\}}},\quad
k\geq 2\end{array}\right.$

where :

  * -

We recall Notation 2.2 for $(x_{(1)},\ldots,x_{(k)})$

  * -

Notation $\sum_{\\{y_{1},\ldots,y_{n}\\}\subset\\{x_{1},\ldots,x_{k-1}\\}}$
stands for the sum over all subsets $\\{y_{1},\ldots,y_{n}\\}$ of cardinal $n$
of $\\{x_{1},\ldots,x_{k-1}\\}$

  * -

$\lambda_{t}(\varpi_{\\{y_{1},\ldots,y_{n}\\}})$ is given by (4.4) in
Proposition 4.4.

###### Proof.

First by [3, 4, 9] the system of SDEs (4.5) admits a unique solution which is
a Hawkes process (we refer to [9] for more details on the construction with
pathwise uniqueness). Fix $T>0$. Then for $M>\mu$, we set :

$\Omega^{M}:=\left\\{\sup_{t\in[0,T]}\lambda_{t}\leq M\right\\}\subset\Omega.$

By Markov’s inequality

$\mathbb{P}[\Omega\setminus\Omega^{M}]\leq\frac{\mathbb{E}\left[\sup_{t\in[0,T]}\lambda_{t}\right]}{M}\leq\frac{\mu+\|\Phi\|_{\infty}\mathbb{E}\left[H_{T}\right]}{M}\leq\frac{\mu+\|\Phi\|_{\infty}\|\Phi\|_{1}(1-\|\Phi\|_{1})^{-1}}{M}.$

Letting $\bar{\Omega}:=\lim_{M\to+\infty}[\Omega\setminus\Omega^{M}]$ where
the limit is understood as an decreasing sequence of sets,
$\mathbb{P}[\bar{\Omega}]=0$.

Fix $M\geq\mu$ and set $(H^{M},\lambda^{M})$ the unique solution to

$\left\\{\begin{array}[]{l}H_{t}^{M}=\int_{(0,t]\times[0,M]}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\\\
\\\
\lambda_{t}^{M}=\mu+\int_{(0,t)\times\mathbb{R}_{+}}\Phi(t-s)dH_{s}^{M},\quad
t\in[0,T].\end{array}\right.$ (4.7)

By construction $H^{M}$ is a counting process with intensity $\lambda\wedge M$
and $(H^{M},\lambda^{M})=(H,\lambda)$ on $\Omega^{M}$ by uniqueness of the
solution to the SDE. Hence by Theorem 3.15, $H_{T}^{M}$ admits a pseudo-
chaotic expansion with respect to $N^{T,M}$ and

$H_{T}^{M}=\sum_{k=1}^{+\infty}\int_{([0,T]\times[0,M])^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),$ (4.8)
$c_{k}(x_{1},\ldots,x_{k}):=\mathbb{E}\left[L^{T,M}D_{(x_{(1)},\ldots,x_{(k-1)})}^{k-1}\textbf{1}_{\\{\theta_{(k)}\leq\lambda_{(t_{k})}\\}}\right],\quad\forall(x_{1},\ldots,x_{k})\in([0,T]\times[0,M])^{k},$
(4.9)

with the ordering convention of Notation 2.2. We should mention that the only
dependency on $M$ in the coefficients $c_{k}$ is simply the domain of the
variables $(x_{1},\ldots,x_{k})$ in $([0,T]\times[0,M])^{k}$. For such $k$ and
$(x_{1},\ldots,x_{k})$ (where for simplicity we assume that
$(x_{(1)},\ldots,x_{(k)})=(x_{1},\ldots,x_{k})$) we have using Proposition 2.5
that for $\omega\in\Omega^{M}$,

$\displaystyle
L^{T,M}(\omega)(D_{(x_{1},\ldots,x_{k-1})}^{k-1}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}\\}})(\omega)$
$\displaystyle=L^{T,M}(\omega)\sum_{J\subset\\{1,\cdots,k-1\\}}(-1)^{k-1-|J|}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\omega+\sum_{j\in
J}\delta_{x_{j}})\\}},$

where the sum is over all subsets $J$ of $\\{1,\cdots,k-1\\}$ including the
empty set which is of cardinal $0$. Hence,

$\displaystyle
L^{T,M}(\omega)(D_{(x_{1},\ldots,x_{k-1})}^{k-1}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}\\}})(\omega)$
$\displaystyle=\exp(MT)\textbf{1}_{\\{(N([0,T]\times[0,M])(\omega))=0\\}}\sum_{J\subset\\{1,\cdots,k-1\\}}(-1)^{k-1-|J|}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\omega+\sum_{j\in
J}\delta_{x_{j}})\\}}$
$\displaystyle=\exp(MT)\textbf{1}_{\\{\omega([0,T]\times[0,M])=0\\}}\sum_{J\subset\\{1,\cdots,k-1\\}}(-1)^{k-1-|J|}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\omega+\sum_{j\in
J}\delta_{x_{j}})\\}}$
$\displaystyle=\exp(MT)\textbf{1}_{\\{\omega([0,T]\times[0,M])=0\\}}\sum_{J\subset\\{1,\cdots,k-1\\}}(-1)^{k-1-|J|}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\sum_{j\in
J}\delta_{x_{j}})\\}}.$

Recall that
$\mathbb{E}[\exp(MT)\textbf{1}_{\\{\omega([0,T]\times[0,M])=0\\}}]=1$. In
other words the effect of $L^{T,M}$ is to freeze the evaluation of the
intensity process $\lambda$ on a specific outcome given by the atoms
$(x_{1},\ldots,x_{k-1})$. Taking the expectation and reorganizing the sum
above we get

$\displaystyle c_{k}(x_{1},\ldots,x_{k})$
$\displaystyle=\mathbb{E}\left[L^{T,M}D_{(x_{1},\ldots,x_{k-1})}^{k-1}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}\\}}\right]$
$\displaystyle=\sum_{J\subset\\{1,\cdots,k-1\\}}(-1)^{k-1-|J|}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\sum_{j\in
J}\delta_{x_{j}})\\}}$
$\displaystyle=(-1)^{k-1}\textbf{1}_{\\{\theta_{k}\leq\mu\\}}+\sum_{n=1}^{k-1}\sum_{\\{y_{1},\ldots,y_{n}\\}\subset\\{x_{1},\ldots,x_{k-1}\\}}(-1)^{k-1-n}\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\varpi_{\\{y_{1},\ldots,y_{n}\\}}\\}}$
$\displaystyle=(-1)^{k-1}\textbf{1}_{\\{\theta_{k}\leq\mu\\}}+\sum_{n=1}^{k-1}\sum_{\\{y_{1},\ldots,y_{n}\\}\subset\\{x_{1},\ldots,x_{k-1}\\}}(-1)^{k-1-n}\textbf{1}_{\\{\theta_{k}\leq
a_{k}^{\\{y_{1},\ldots,y_{n}\\}}\\}}.$

Note that in each term
$\textbf{1}_{\\{\theta_{k}\leq\lambda_{t_{k}}(\sum_{j\in
J}\delta_{x_{j}})\\}}$, $\sum_{j\in J}\delta_{x_{j}}$ is deterministic and
$\lambda_{t_{k}}(\sum_{j\in J}\delta_{x_{j}})$ is explicitly given by the
triangular system in Notation 4.4. For $k=1$, the previous expression just
reduces to

$c_{1}(x_{1})=\mathbb{E}\left[L^{T,M}\textbf{1}_{\\{\theta_{1}\leq\lambda_{t_{1}}\\}}\right]=\textbf{1}_{\\{\theta_{1}\leq\mu\\}}.$

Finally, as

$H_{T}(\omega)=H_{T}^{M}(\omega),\quad\textrm{ on }\Omega^{M},$

for any $k\geq 1$

$\displaystyle\int_{([0,T]\times[0,M])^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k})$
$\displaystyle=\int_{\mathbb{X}^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k}),\;\textrm{ on }\Omega^{M}$

and thus the expansion holds true on $\Omega\setminus\bar{\Omega}$ and
$\mathbb{P}[\bar{\Omega}]=0$. ∎

######  Remark 4.6.

The boundedness assumption on $\Phi$ in Theorem 4.5 is not sharpe and can be
replaced with any assumption ensuring that for any $T>0$,
$\mathbb{E}[\sup_{t\in[0,T]}\lambda_{t}]<+\infty$.

######  Remark 4.7.

To the price of cumbersome notations, the previous result can be extended to
non-linear Hawkes processes; that is counting processes $H$ with intensity
process of the form

$\left\\{\begin{array}[]{l}H_{t}=\int_{(0,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}N(ds,d\theta),\\\
\\\
\lambda_{t}=h\left(\mu+\int_{(0,t)\times\mathbb{R}_{+}}\Phi(t-s)dH_{s}\right),\quad
t\in[0,T].\end{array}\right.$ (4.10)

where $h:\mathbb{R}\to\mathbb{R}_{+}$, $\Phi:\mathbb{R}_{+}\to\mathbb{R}$ and
$\|h\|_{1}\|\Phi\|_{1}<1$. Indeed, when computing the coefficients in the
expansion, the Poisson measure $N$ is cancelled and involves evaluations of
the intensity process at a specific configurations of the form
$\varpi_{\\{y_{1},\ldots,y_{n}\\}}$. This evaluation can be done by a
straightforward extension of Proposition 4.4 for a non-linear Hawkes process;
in other words for both linear or non-linear process the intensity process
$\lambda$ is a deterministic function of the fixed configuration of the form
$\varpi_{\\{y_{1},\ldots,y_{n}\\}}$.

######  Discussion 4.8.

We would like to comment on the advantage of the pseudo-chaotic expansion
compared to the usual one for the value $H_{T}$ of a linear Hawkes process at
any time $T$. Recall the two decompositions

$\displaystyle H_{T}$
$\displaystyle=\mathbb{E}[H_{T}]+\sum_{j=1}^{+\infty}\frac{1}{j!}I_{j}(f_{j}^{H_{T}}),\quad\textrm{
with }\quad
f_{j}^{H_{T}}(x_{1},\cdots,x_{j})=\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}H_{T}\right]$
$\displaystyle=\sum_{k=1}^{+\infty}\int_{\mathbb{X}^{k}}\frac{1}{k!}c_{k}(x_{1},\ldots,x_{k})N(dx_{1})\cdots
N(dx_{k})$

For the chaotic expansion, in order to determine each coefficient $f_{j}$ one
has to compute
$f_{j}^{H_{T}}(x_{1},\cdots,x_{j})=\mathbb{E}\left[D_{(x_{1},\cdots,x_{j})}^{j}H_{T}\right]$
which turns out to be quite implicit for a general $\Phi$ kernel. Indeed,
already for the first coefficient, using Relation (3.12) we have

$\displaystyle f_{j}^{H_{T}}(x_{1})$
$\displaystyle=\mathbb{E}\left[D_{x_{1}}H_{T}\right]$
$\displaystyle=\mathbb{E}\left[\textbf{1}_{\\{\theta_{1}\leq\lambda_{t_{1}}\\}}\right]+\int_{(t_{1},T]\times\mathbb{R}_{+}}\mathbb{E}\left[D_{x_{1}}\textbf{1}_{\\{\theta\leq\lambda_{s}\\}}\right]d\theta
ds$
$\displaystyle=\mathbb{P}\left[\theta_{1}\leq\lambda_{t_{1}}\right]+\int_{(t_{1},T]}\mathbb{E}\left[D_{x_{1}}\lambda_{s}\right]ds.$

The quantity $\int_{(t_{1},T]}\mathbb{E}\left[D_{x_{1}}\lambda_{s}\right]ds$
has been computed in [8] however a closed form expression for
$\mathbb{P}\left[\theta_{1}\leq\lambda_{t_{1}}\right]$ for any kernel $\Phi$
satisfying Assumption 4.1 is unknown to the authors.

In contradistinction, Theorem 4.5 gives an explicit expression for the
coefficients $c_{k}$. In that sense, the pseudo-chaotic expansion (4.6) is an
exact representation and an explicit solution to the Hawkes equation
formulation as given in Definition 4.3.

##  5 The pseudo-chaotic expansion and the Hawkes equation

The aim of this section is to investigate further the link between a
decomposition of the form (4.8) that we named pseudo-chaotic expansion and the
characterization of a Hawkes process as in Definition 4.3. First, let us
emphasize that both the standard chaotic expansion and the pseudo-chaotic
expansion characterize a given random variable and not a stochastic process.
For instance in Theorem 4.5, the coefficients $c_{k}$ for the expansion of
$H_{T}$ depend on the time $T$. In this section, we consider once again the
linear Hawkes process, which is essentially described as a counting process
with a specific stochastic intensity like in (1.1), and we adopt a different
point of view based on population dynamics and branching representation as in
[7] or [2]. Inspired by this branching representation, we build in Theorem 5.2
below a stochastic process via its pseudo-chaotic expansion which is an
integer-valued piecewise-constant and non-decreasing process with the specific
intensity form of a Hawkes process. Nevertheless, although this stochastic
process satisfies the stochastic self-exciting intensity equation which
determines a Hawkes process, it fails to be a counting process as it may
exhibit jumps larger than one. This leaves open further developments for
studying the pseudo-chaotic expansion of processes; we refer to Discussion
5.3.

###  5.1 A pseudo-chaotic expansion and branching representation

Throughout this section we will make use of classical stochastic analysis
tools hence we describe elements of $\mathbb{X}$ as $(t,\theta)$ instead of
$x$ for avoiding any confusion. The branching representation viewpoint
consists in counting the number of individuals in generation $n$, where
generation $1$ corresponds of the migrants. We therefore define a series of
counting processes $X_{t}^{(n)}$ (where $n$ stands for the generation) as
follows

######  Definition-Proposition 5.1.

(i) Let for any $t\geq 0$

$X_{t}^{(1)}:=\int_{(0,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}N(dv_{1},d\theta_{1}),$
(5.1)

and for $n\geq 2$,

$X_{t}^{(n)}:=\int_{(0,t]\times\mathbb{R}_{+}}\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n}).$ (5.2)

We set in addition

$X_{t}:=\sum_{n=1}^{+\infty}X_{t}^{(n)},$ (5.3)

where the series converges uniformly (in $t$) on compact sets; that is for any
$T>0$,

$\lim_{p\to+\infty}\mathbb{E}\left[\sup_{t\in[0,T]}\left|X_{t}-\sum_{n=1}^{p}X_{t}^{(n)}\right|\right]=0.$

(ii) We set the $\mathbb{F}^{X}$-predictable process

$\ell_{t}:=\mu+\int_{(0,t)}\Phi(t-r)dX_{r}=\mu+\sum_{n=1}^{+\infty}\int_{(0,t)}\Phi(t-r)dX_{r}^{(n)},\quad
t\geq 0$ (5.4)

where $\mathbb{F}^{X}:=(\mathcal{F}^{X}_{t})_{t\geq 0}$, with
$\mathcal{F}^{X}_{t}:=\sigma(X_{s},\;s\leq t)$.

The proof of the convergence of the series (5.3) is postponed to Section 5.2.
The resulting process $X$ aims at counting the number of individuals in the
population, while the predictable process $\ell$ is the candidate to be the
self-exciting intensity of the process $X$. This intensity reads as follows

$\displaystyle\ell_{t}$ $\displaystyle=\mu+\int_{(0,t)}\Phi(t-r)dX_{r}$
$\displaystyle=\mu+\int_{(0,t)\times\mathbb{R}_{+}}\Phi(t-v_{1})\textbf{1}_{\\{\theta_{1}\leq\mu\\}}N(dv_{1},d\theta_{1})$
$\displaystyle+\sum_{n=2}^{+\infty}\int_{(0,t]\times\mathbb{R}_{+}}\Phi(t-v_{n})\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n})$
$\displaystyle=\mu+\sum_{n=1}^{+\infty}\int_{(0,t]\times\mathbb{R}_{+}}\Phi(t-v_{n})\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n}).$

The main result of this section is stated below. Its proof based on Lemmata
5.7 and 5.8 is postponed to Section 5.3.

######  Theorem 5.2.

Let $\mu>0$ and $\Phi$ satisfying Assumption 4.1. Recall the stochastic
process $X$ and $\ell$ defined (in Definition-Proposition 5.1). We define the
process $M:=(M_{t})_{t\geq 0}$ by

$M_{t}:=X_{t}-\int_{(0,t)}\ell_{u}du,\quad t\geq 0.$ (5.5)

Then $X$ is a $\mathbb{N}$-valued non-decreasing process piecewise constant
with predictable intensity process $\ell$, in the sense that process $M$ is a
$\mathbb{F}^{N}$-martingale (and so a $\mathbb{F}^{H}$-martingale as $M$ is
$\mathbb{F}^{H}$-adapted and
$\mathcal{F}^{H}_{\cdot}\subset\mathcal{F}^{N}_{\cdot}$).

######  Discussion 5.3.

In other words $X$ would be a Hawkes process if it were a counting process,
unless it has the same expectation of a Hawkes process. Indeed

  1. 1.

some atoms generates simultaneous jumps : any atom $(t_{0},\theta_{0})$ of $N$
with $\theta_{0}\leq\mu$ will generate a jump for $X^{(1)}$ and for all
$X^{(n)}$ who have $t_{0}$ as an ancestor so $X_{t_{0}}-X_{{t_{0}}-}$ may be
larger than one.

  2. 2.

some atoms are ignored by $X$ : by construction any atom $(t_{0},\theta_{0})$
of $N$ with $\theta>\|\Phi\|_{\infty}$ is ignored by the process, whereas the
area of decision $\textbf{1}_{\\{\theta\leq\lambda_{t}\\}}$ is unbounded in
the $\theta$-variable for a linear Hawkes process.

This example leads to a question. More specifically, the more intricate
structure of the coefficients in Theorem 5.2 for the pseudo-chaotic expansion
of a Hawkes process as a sum and differences of indicator functions suggests a
necessary algebraic structure with respect to the time variable on the
coefficients of the expansion to guarantee the counting-feature of the
process. We leave this issue for future research.

Before handling in Section 5.3 the proof of Theorem 5.2, we start with some
useful technical lemmata.

###  5.2 Technical results and proofs

######  Lemma 5.4.

Let $f$ in $L_{1}(\mathbb{R}_{+};dt)$. For any $n\in\mathbb{N}$ with $n\geq
3$, and for any $0\leq s\leq t$,

$\int_{s}^{t}\int_{s}^{u}\Phi_{n-1}(t-r)f(r)drdu=\int_{s}^{t}\int_{s}^{v_{n}}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\prod_{i=2}^{n}\Phi(v_{i}-v_{i-1})f({v_{1}})dv_{1}\cdots
dv_{n}.$ (5.6)

###### Proof.

For $g$ a mapping, let $\mathfrak{F}(g)$ the Fourier transform of $g$. Fix
$s\geq 0$ and $n\geq 3$. Let

$F(u):=\int_{s}^{u}\Phi_{n-1}(u-r)f(r)dr,\;u\geq s$

so that $F=\Phi_{n-1}\ast\tilde{f}$, with
$\tilde{f}(v):=f(v)\textbf{1}_{\\{v\geq s\\}}$. We have that

$\mathfrak{F}(F)=\mathfrak{F}(\Phi_{n-1})\mathfrak{F}(\tilde{f})=\mathfrak{F}(\Phi)^{n-1}\mathfrak{F}(\tilde{f}).$

In addition by definition of mappings $\Phi_{i}$ (see Relation (4.1)), each
$\Phi_{i}$ is the $i$th convolution of $\Phi$ with itself; hence
$\mathfrak{F}(\Phi_{n-1})=(\mathfrak{F}(\Phi))^{n-1}$. Let :

$G(u):=\int_{s}^{u}\Phi(u-v_{n-1})\int_{s}^{v_{n-1}}\Phi(v_{n-1}-v_{n-2})\cdots\int_{s}^{v_{2}}\Phi(v_{2}-v_{1})f({v_{1}})dv_{1}\cdots
dv_{n-1},$

we immediately get that
$\mathfrak{F}(G)=(\mathfrak{F}(\Phi))^{n-2}\mathfrak{F}(\Phi)\mathfrak{F}(\tilde{f})=(\mathfrak{F}(\Phi))^{n-1}\mathfrak{F}(\tilde{f})=\mathfrak{F}(F)$.
Using the inverse Fourier transform (on the left) we get that $F(u)=G(u)$ for
a.e. $u$ leading to $\int_{s}^{t}F(u)du=\int_{s}^{t}G(u)du$ which is Relation
(5.6). ∎

Lemma 5.4 allows one to prove Proposition 5.1, namely to prove that the series
$X_{t}=\sum_{n=1}^{+\infty}X_{t}^{(n)}$ converges uniformly (in $t$) on
compact sets; that is for any $T>0$,

$\lim_{p\to+\infty}\mathbb{E}\left[\sup_{t\in[0,T]}\left|X_{t}-\sum_{n=1}^{p}X_{t}^{(n)}\right|\right]=0.$

###### Proof.

Set $T>0$. For $p\geq 2$, let $S_{p}:=\sum_{n=1}^{p}X^{(n)}$. As each
$X^{(n)}$ process is non-negative as a counting process we have that :

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|X_{t}-S_{p}(t)\right|\right]$
$\displaystyle=\mathbb{E}\left[\sup_{t\in[0,T]}\left|\sum_{n=p+1}^{+\infty}X^{(n)}_{t}\right|\right]$
$\displaystyle=\sum_{n=p+1}^{+\infty}\mathbb{E}\left[X^{(n)}_{T}\right]$
$\displaystyle=\mu\sum_{n=p+1}^{+\infty}\int_{0}^{T}\int_{0}^{v_{n}}\cdots\int_{0}^{v_{2}}\prod_{i=2}^{n}\Phi(v_{i}-v_{i-1})dv_{1}\cdots
dv_{n}$
$\displaystyle=\mu\sum_{n=p+1}^{+\infty}\int_{0}^{T}\int_{0}^{t}\Phi_{n-1}(T-r)drdt,\quad\textrm{
by Lemma \ref{lemma:magic}}$ $\displaystyle\leq\mu
T\sum_{n=p}^{+\infty}\|\Phi_{n}\|_{1}=\mu
T\frac{\|\Phi\|_{1}^{p}}{1-\|\Phi\|_{1}}\underset{p\to+\infty}{\longrightarrow}0,\quad\textrm{
by Proposition \ref{prop:Phin}}.$

∎

######  Lemma 5.5.

For $n\geq 3$, $0\leq s\leq u$ we set

$\displaystyle Q(n,u)$
$\displaystyle:=\mathbb{E}_{s-}\left[\int_{(0,u]}\Phi(u-r)dX_{r}^{(n-1)}\right]$
$\displaystyle=\mathbb{E}_{s-}\left[\int_{(0,u]\times\mathbb{R}_{+}}\Phi(u-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})\right].$

We have

$\displaystyle Q(n,u)$ $\displaystyle=h_{u}^{s,(n-1)}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{u}\cdots\int_{s}^{v_{n-i+1}^{*}}\Phi(u-v_{n-1})\prod_{k=n-i+1}^{n-1}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}$
$\displaystyle+\mu\int_{s}^{u}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\Phi(u-v_{n-1})\prod_{k=2}^{n-1}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1},$ (5.7)

where $v_{n-i+1}^{*}:=v_{n-i+1}$ for $i\neq 1$ and $v_{n-i+1}^{*}:=u$ for
$i=1$. An explicit computation gives that Relation (5.5) is valid for $n=2$
using Convention 2.1.

###### Proof.

Using Fubini’s theorem777here Fubini’s theorem is used pathwise as the
integral against $N$ are finite-a.e. we have

$\displaystyle Q(n,u)$
$\displaystyle=\int_{(0,s]\times\mathbb{R}_{+}}\Phi(u-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})$
$\displaystyle+\int_{s}^{u}\Phi(u-v_{n-1})\mathbb{E}_{s-}\left[\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})\right]dv_{n-1}$
$\displaystyle=\int_{(0,s]\times\mathbb{R}_{+}}\Phi(u-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})$
$\displaystyle+\int_{s}^{u}\Phi(u-v_{n-1})\mathbb{E}_{s-}\left[\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\Phi(v_{n-1}-v_{n-2})\int_{(0,v_{n-2}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\right.$
$\displaystyle\left.\hskip
100.00015pt\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-2}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})\right]dv_{n-1}$
$\displaystyle=\int_{(0,s]\times\mathbb{R}_{+}}\Phi(u-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})$
$\displaystyle+\int_{s}^{u}\Phi(u-v_{n-1})Q(n-1,v_{n-1})dv_{n-1}$
$\displaystyle=h_{u}^{s,(n-1)}+\int_{s}^{u}\Phi(u-v_{n-1})Q(n-1,v_{n-1})dv_{n-1}.$

So we have proved that

$Q(n,u)=h_{u}^{s,(n-1)}+\int_{s}^{u}\Phi(u-v_{n-1})Q(n-1,v_{n-1})dv_{n-1}.$

We then deduce by induction that

$\displaystyle Q(n,u)$ $\displaystyle=h_{u}^{s,(n-1)}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{u}\cdots\int_{s}^{v_{n-i+1}^{*}}\Phi(u-v_{n-1})\prod_{k=n-i+1}^{n-1}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}$
$\displaystyle+\mu\int_{s}^{u}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\Phi(u-v_{n-1})\prod_{k=2}^{n-1}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}.$

Indeed, assuming the previous relation is true for $Q(n,u)$ for a given $n\geq
3$ and for any $u$; we have

$\displaystyle Q(n+1,u)$
$\displaystyle=h_{u}^{s,(n)}+\int_{s}^{u}\Phi(u-v_{n})Q(n,v_{n})dv_{n}$
$\displaystyle=h_{u}^{s,(n)}+\int_{s}^{u}\Phi(u-v_{n})h_{v_{n}}^{s,(n-1)}dv_{n}$
$\displaystyle+\int_{s}^{u}\Phi(u-v_{n})\left[\sum_{i=1}^{n-2}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}}\Phi(v_{n}-v_{n-1})\prod_{k=n-i+1}^{n-1}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}\right]dv_{n}$
$\displaystyle+\mu\int_{s}^{u}\Phi(u-v_{n})\int_{s}^{v_{n-1}}\Phi(v_{n}-v_{n-1})\prod_{k=2}^{n-1}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}dv_{n}$
$\displaystyle=h_{u}^{s,(n)}+\int_{s}^{u}\Phi(u-v_{n})h_{v_{n}}^{s,(n-1)}dv_{n}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{u}\Phi(u-v_{n})\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n}-i+1}\prod_{k=n-i+1}^{n}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}dv_{n}$
$\displaystyle+\mu\int_{s}^{u}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\Phi(u-v_{n})\prod_{k=2}^{n}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}dv_{n}$ $\displaystyle=h_{u}^{s,(n)}$
$\displaystyle+\sum_{i=0}^{n-2}\int_{s}^{u}\Phi(u-v_{n})\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}}\Phi(v_{n}-v_{n-1})\prod_{k=n-i+1}^{n}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}dv_{n}$
$\displaystyle+\mu\int_{s}^{u}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\Phi(u-v_{n})\prod_{k=2}^{n}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}dv_{n},$

where for $i=0$ we use Convention 2.1. Hence

$\displaystyle Q(n+1,u)$ $\displaystyle=h_{u}^{s,(n)}$
$\displaystyle+\sum_{j=1}^{(n+1)-2}\int_{s}^{u}\Phi(u-v_{n})\int_{s}^{v_{n}}\cdots\int_{s}^{v_{(n+1)-j+1}^{*}}\prod_{k=(n+1)-j+1}^{n+1}\Phi(v_{k}-v_{k-1})h_{v_{n+1-j}}^{s,((n+1)-j-1)}dv_{(n+1)-j}\ldots
dv_{n-1}dv_{n}$
$\displaystyle+\mu\int_{s}^{u}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\Phi(u-v_{n})\prod_{k=2}^{(n+1)-1}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{(n+1)-1},$

which gives the result. ∎

###  5.3 Proof of Theorem 5.2

The proof consists in showing that the process $M$ defined by (5.5) is a
$\mathbb{F}^{N}$-martingale, that is for any $0\leq s\leq t$,

$\mathbb{E}_{s-}\left[X_{t}-X_{s}\right]=\int_{s}^{t}\mathbb{E}_{s-}[\ell_{r}]dr,$
(5.8)

where for simplicity
$\mathbb{E}_{s-}[\cdot]:=\mathbb{E}[\cdot|\mathcal{F}_{s-}^{N}]$. This result
is a direct consequence of Lemma 5.7 and 5.8 below in which we compute both
terms in (5.8). To this end we introduce the following notation

######  Notation 5.6.

Let $s\geq 0$, $v\geq s$ and $n\geq 1$, we set

$h_{v}^{s}:=\sum_{n=1}^{+\infty}h_{v}^{s,(n)},$

with

$\displaystyle h_{v}^{s,(n)}$
$\displaystyle:=\int_{(0,s)}\Phi(v-v_{n})dX_{v_{n}}^{(n)}$
$\displaystyle=\int_{(0,s)\times\mathbb{R}_{+}}\Phi(v-v_{n})\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n}).$

Lemma 5.7 first compute the left-hand side of in (5.8).

######  Lemma 5.7.

For any $0\leq s\leq t$ we have that

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}\right]=\int_{s}^{t}(\mu+h_{u}^{s})du+\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu.$
(5.9)

###### Proof.

We have :

$\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(1)}\right]=\mathbb{E}_{s-}\left[\int_{(s,t]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}N(dv_{1},d\theta_{1})\right]=\mu(t-s).$
(5.10)

Let $n\geq 2$.

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(n)}\right]$
$\displaystyle=\mathbb{E}_{s-}\left[\int_{(s,t]\times\mathbb{R}_{+}}\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n})\right]$
$\displaystyle=\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(0,v_{n}]\times\mathbb{R}_{+}}\Phi(v_{n}-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})\right]dv_{n}$
$\displaystyle=\int_{s}^{t}Q(n,v_{n})dv_{n}.$

Hence, by Lemma 5.5,

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(n)}\right]$
$\displaystyle=\int_{s}^{t}h_{v_{n}}^{s,(n-1)}dv_{n}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}}\prod_{k=n-i+1}^{n}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n}$
$\displaystyle+\mu\int_{s}^{t}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{2}}\prod_{k=2}^{n}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n}.$ (5.11)

Using Lemma 5.4,

$\displaystyle\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}}\prod_{k=n-i+1}^{n}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n}$
$\displaystyle=\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i}(u-r)h_{r}^{s,(n-i-1)}drdu,$

and

$\mu\int_{s}^{t}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{2}}\prod_{k=2}^{n}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n}=\mu\int_{s}^{t}\int_{s}^{u}\Phi_{n-1}(u-r)drdu.$

Plugging back these expressions in (5.3) we get

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(n)}\right]$
$\displaystyle=\int_{s}^{t}h_{u}^{s,(n-1)}du+\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i}(u-r)h_{r}^{s,(n-i-1)}drdu+\mu\int_{s}^{t}\int_{s}^{u}\Phi_{n-1}(u-r)drdu.$
(5.12)

We now sum the previous quantity over $n\geq 2$. The main term to be treated
is the second one that we treat separately below. Note also that using
Convention 2.1 for $n=2$ we get

$\displaystyle\sum_{n=2}^{+\infty}\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i}(u-r)h_{r}^{s,(n-i-1)}drdu$
$\displaystyle=\sum_{n=3}^{+\infty}\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i}(u-r)h_{r}^{s,(n-i-1)}drdu$
$\displaystyle=\sum_{n=3}^{+\infty}\sum_{j=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{n-j-1}(u-r)h_{r}^{s,(j)}drdu$
$\displaystyle=\sum_{j=1}^{+\infty}\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(j)}\left(\sum_{n=3}^{+\infty}\textbf{1}_{\\{j\leq
n-2\\}}\Phi_{n-j-1}(u-r)\right)drdu$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(1)}\left(\sum_{n=3}^{+\infty}\Phi_{n-2}(u-r)\right)drdu+\sum_{j=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(j)}\left(\sum_{n=j+2}^{+\infty}\Phi_{n-j-1}(u-r)\right)drdu$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(1)}\left(\sum_{k=1}^{+\infty}\Phi_{k}(u-r)\right)drdu+\sum_{j=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(j)}\left(\sum_{k=1}^{+\infty}\Phi_{k}(u-r)\right)drdu$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(1)}\Psi(u-r)drdu+\sum_{j=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(j)}\Psi(u-r)drdu$
$\displaystyle=\sum_{j=1}^{+\infty}\int_{s}^{t}\int_{s}^{u}h_{r}^{s,(j)}\Psi(u-r)drdu$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}\Psi(u-r)h_{r}^{s}drdu.$ (5.13)

With these computations at hand, Relations (5.3) and (5.3) lead to

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}\right]$
$\displaystyle=\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(n)}\right]+\sum_{n=2}^{+\infty}\mathbb{E}_{s-}\left[\int_{(s,t]}dX_{r}^{(n)}\right]$
$\displaystyle=\int_{s}^{t}\mu
du+\sum_{n=2}^{+\infty}\int_{s}^{t}h_{u}^{s,(n-1)}du+\sum_{n=2}^{+\infty}\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i}(u-r)h_{r}^{s,(n-i-1)}drdu+\mu\sum_{n=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}\Phi_{n-1}(u-r)drdu$
$\displaystyle=\int_{s}^{t}(\mu+h_{u}^{s})du+\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu,$

which concludes the proof. ∎

We now compute the right-hand side in (5.8).

######  Lemma 5.8.

For any $0\leq s\leq t$ we have that :

$\displaystyle\mathbb{E}_{s-}\left[\int_{s}^{t}\ell_{r}dr\right]=\int_{s}^{t}(\mu+h_{u}^{s})du+\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu.$
(5.14)

###### Proof.

The proof is rather similar to the one of Lemma 5.7, we provide a proof for
the sake of completeness.
Let $0\leq s\leq r\leq t$. Recall Notation 5.6. We have

$\displaystyle\mathbb{E}_{s-}\left[\ell_{r}\right]$
$\displaystyle=\mathbb{E}_{s-}\left[\mu+\int_{(0,r)}\Phi(r-u)dX_{u}\right]$
$\displaystyle=\mu+\sum_{n=1}^{+\infty}\mathbb{E}_{s-}\left[\int_{(0,r)}\Phi(r-u)dX_{u}^{(n)}\right]$
$\displaystyle=\mu+\sum_{n=1}^{+\infty}\int_{(0,s)}\Phi(r-u)dX_{u}^{(n)}+\sum_{n=1}^{+\infty}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]$
$\displaystyle=\mu+h_{r}^{s}+\sum_{n=1}^{+\infty}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right].$
(5.15)

Let $n\geq 2$,

$\displaystyle\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]$
$\displaystyle=\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-v_{n})\int_{(0,v_{n}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n},d\theta_{n})\right]$
$\displaystyle=\int_{s}^{r}\Phi(r-v_{n})\mathbb{E}_{s-}\left[\int_{(0,v_{n}]\times\mathbb{R}_{+}}\Phi(v_{n}-v_{n-1})\int_{(0,v_{n-1}]\times\mathbb{R}_{+}}\cdots\int_{(0,v_{2}]\times\mathbb{R}_{+}}\right.$
$\displaystyle\left.\hskip
50.00008pt\textbf{1}_{\\{\theta_{1}\leq\mu\\}}\prod_{i=2}^{n-1}\textbf{1}_{\\{\theta_{i}\leq\Phi(v_{i}-v_{i-1})\\}}N(dv_{1},d\theta_{1})\cdots
N(dv_{n-1},d\theta_{n-1})\right]dv_{n}$
$\displaystyle=\int_{s}^{r}\Phi(r-v_{n})Q(n,v_{n})dv_{n}$
$\displaystyle=\int_{s}^{r}\Phi(r-v_{n})h_{v_{n}}^{s,(n-1)}dv_{n}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{r}\Phi(r-v_{n})\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}^{*}}\prod_{k=n-i+1}^{n}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}dv_{n}$
$\displaystyle+\mu\int_{s}^{r}\Phi(r-v_{n})\int_{s}^{v_{n}}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\prod_{k=2}^{n}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}dv_{n}$

where the last equality follows from Lemma 5.5. Integrating the previous
expression in $r$ on $(s,t]$ and using Lemma 5.4 one gets

$\displaystyle\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]dr$
$\displaystyle=\int_{s}^{t}\int_{s}^{v_{n}}\Phi(v_{n}-v_{n+1})h_{v_{n+1}}^{s,(n-1)}dv_{n+1}dv_{n}$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{v_{n+1}}\int_{s}^{v_{n}}\cdots\int_{s}^{v_{n-i+1}^{*}}\prod_{k=n-i+1}^{n+1}\Phi(v_{k}-v_{k-1})h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots
dv_{n-1}dv_{n}dv_{n+1}$
$\displaystyle+\mu\int_{s}^{t}\int_{s}^{v_{n+1}}\int_{s}^{v_{n}}\int_{s}^{v_{n-1}}\cdots\int_{s}^{v_{2}}\prod_{k=2}^{n+1}\Phi(v_{k}-v_{k-1})dv_{1}\ldots
dv_{n-1}dv_{n}dv_{n+1}$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}\Phi(u-r)h_{r}^{s,(n-1)}drdu$
$\displaystyle+\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i+1}(u-r)h_{r}^{s,(n-i-1)}drdu$
$\displaystyle+\mu\int_{s}^{t}\int_{s}^{u}\Phi_{n}(u-r)drdu$

Using the same computations than (5.3) we deduce (using Convention 2.1) that

$\displaystyle\sum_{n=1}^{+\infty}\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]dr$
$\displaystyle=\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(1)}\right]+\sum_{n=2}^{+\infty}\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]dr$
$\displaystyle=\mu\int_{s}^{t}\int_{s}^{u}\Phi(r-u)drdu+\sum_{n=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}\Phi(u-r)h_{r}^{s,(n-1)}drdu$
$\displaystyle+\sum_{n=3}^{+\infty}\sum_{i=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{i+1}(u-r)h_{r}^{s,(n-i-1)}drdu+\sum_{n=2}^{+\infty}\mu\int_{s}^{t}\int_{s}^{u}\Phi_{n}(u-r)drdu$
$\displaystyle=\mu\int_{s}^{t}\int_{s}^{u}\Psi(u-r)drdu$
$\displaystyle+\sum_{n=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}\Phi(u-r)h_{r}^{s,(n-1)}drdu$
$\displaystyle+\sum_{n=3}^{+\infty}\sum_{j=2}^{(n+1)-2}\int_{s}^{t}\int_{s}^{u}\Phi_{j}(u-r)h_{r}^{s,(n+1-j-1)}drdu$
$\displaystyle=\mu\int_{s}^{t}\int_{s}^{u}\Psi(u-r)drdu+\sum_{n=2}^{+\infty}\int_{s}^{t}\int_{s}^{u}\Phi(u-r)h_{r}^{s,(n-1)}drdu+\sum_{n=2}^{+\infty}\sum_{j=2}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{j}(u-r)h_{r}^{s,(n-j-1)}drdu$
$\displaystyle=\mu\int_{s}^{t}\int_{s}^{u}\Psi(u-r)drdu+\sum_{n=2}^{+\infty}\sum_{j=1}^{n-2}\int_{s}^{t}\int_{s}^{u}\Phi_{j}(u-r)h_{r}^{s,(n-j-1)}drdu$
$\displaystyle=\mu\int_{s}^{t}\int_{s}^{u}\Psi(u-r)drdu+\int_{s}^{t}\int_{s}^{u}\Psi(u-r)h_{r}^{s}drdu$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu.$

where we have used Relation (5.3). Thus we have proved that

$\displaystyle\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}\right]dr$
$\displaystyle=\sum_{n=1}^{+\infty}\int_{s}^{t}\mathbb{E}_{s-}\left[\int_{(s,r)}\Phi(r-u)dX_{u}^{(n)}\right]dr$
$\displaystyle=\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu.$

Hence coming back to Relation (5.3) we obtain

$\mathbb{E}_{s-}\left[\int_{s}^{t}\ell_{r}dr\right]=\int_{s}^{t}(\mu+h_{u}^{s})du+\int_{s}^{t}\int_{s}^{u}\Psi(u-r)(\mu+h_{r}^{s})drdu.$

∎

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