arXiv:1506.02361v1 [] 8 Jun 2015

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PDE˙Hawkes˙Marie11

Microscopic approach of a time elapsed neural model

Julien Chevallier
Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc
Valrose
06108 Nice Cedex 2, France
julien.chevallier@unice.fr
Marı́a José Cáceres
Departamento de Matemática Aplicada , Universidad de Granada, Campus de Fuentenueva
E-18071 Granada, Spain
caceresg@ugr.es
Marie Doumic
UPMC University of Paris 6, JL Lions Lab., 4 place Jussieu
75005 Paris, France
Patricia Reynaud-Bouret
Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc
Valrose
06108 Nice Cedex 2, France
patricia.reynaud-bouret@unice.fr

The spike trains are the main components of the information processing in the brain.
To model spike trains several point processes have been investigated in the literature. And
more macroscopic approaches have also been studied, using partial differential equation
models. The main aim of the present article is to build a bridge between several point
processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit
real spike trains data and age-structured partial differential equations as introduced by
Pakdaman, Perthame and Salort.
Keywords: Hawkes process; Wold process; renewal equation; neural network
AMS Subject Classification:35F15, 35B10, 92B20, 60G57, 60K15

Introduction
In Neuroscience, the action potentials (spikes) are the main components of the realtime information processing in the brain. Indeed, thanks to the synaptic integration,
the membrane voltage of a neuron depends on the action potentials emitted by some
others, whereas if this membrane potential is sufficiently high, there is production
of action potentials.
1

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To access those phenomena, schematically, one can proceed in two ways: extracellularly record in vivo several neurons, at a same time, and have access to
simultaneous spike trains (only the list of events corresponding to action potentials) or intracellularly record the whole membrane voltage of only one neuron at a
time, being blind to the nearby neurons.
Many people focus on spike trains. Those data are fundamentally random and
can be modelled easily by time point processes, i.e. random countable sets of points
on R+ . Several point processes models have been investigated in the literature, each
of them reproducing different features of the neuronal reality. The easiest model is
the homogeneous Poisson process, which can only reproduce a constant firing rate
for the neuron, but which, in particular, fails to reproduce refractory periodsa . It
is commonly admitted that this model is too poor to be realistic. Indeed, in such a
model, two points or spikes can be arbitrary close as soon as their overall frequency
is respected in average. Another more realistic model is the renewal process 37 ,
where the occurrence of a point or spike depends on the previous occurrence. More
precisely, the distribution of delays between spikes (also called inter-spike intervals,
ISI) is given and a distribution, which provides small weights to small delays, is
able to mimic refractory periods. A deeper statistical analysis has shown that Wold
processes is showing good results, with respect to goodness-of-fit test on real data
sets 38 . Wold processes are point processes for which the next occurrence of a spike
depends on the previous occurrence but also on the previous ISI. From another
point of view, the fact that spike trains are usually non stationary can be easily
modelled by inhomogeneous Poisson processes 43 . All those models do not reflect
one of the main features of spike trains, which is the synaptic integration and there
has been various attempts to catch such phenomenon. One of the main model is the
Hawkes model, which has been introduced in 13 and which has been recently shown
to fit several stationary data 40 . Several studies have been done in similar directions
(see for instance 5 ). More recently a vast interest has been shown to generalized
linear models 36 , with which one can infer functional connectivity and which are
just an exponential variant of Hawkes models.
There has also been several models of the full membrane voltage such as
Hodgkin-Huxley models. It is possible to fit some of those probabilistic stochastic differential equations (SDE) on real voltage data 22 and to use them to estimate
meaningful physiological parameters 18 . However, the lack of simultaneous data
(voltages of different neurons at the same time) prevent these models to be used as
statistical models that can be fitted on network data, to estimate network parameters. A simple SDE model taking synaptic integration into account is the well-known
Integrate-and-Fire (IF) model. Several variations have been proposed to describe
several features of real neural networks such as oscillations 7,8 . In particular, there
exists hybrid IF models including inhomogeneous voltage driven Poisson process 21
that are able to mimic real membrane potential data. However up to our knowledge
a Biologically,

a neuron cannot produce two spikes too closely in time.

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and unlike point processes models, no statistical test have been applied to show
that any of the previous variations of the IF model fit real network data.
Both, SDE and point processes, approaches are microscopic descriptions, where
random noise explains the intrinsic variability. Many authors have argued that there
must be some more macroscopic approach describing huge neural networks as a
whole, using PDE formalism 15,42 . Some authors have already been able to perform
link between PDE approaches as the macroscopic system and SDE approach (in
particular IF models) as the microscopic model 39,30,26 . Another macroscopic point
of view on spike trains is proposed by Pakdaman, Perthame and Salort in a series
of articles 31,32,33 . It uses a nonlinear age-structured equation to describe the spikes
density. Adopting a population view, they aim at studying relaxation to equilibrium or spontaneous periodic oscillations. Their model is justified by a qualitative,
heuristic approach. As many other models, their model shows several qualitative
features such as oscillations that make it quite plausible for real networks, but once
again there is no statistical proof of it, up to our knowledge.
In this context, the main purpose of the present article is to build a bridge between several point processes models that have been proved to statistically fit real
spike trains data and age structured PDE of the type of Pakdaman, Perthame and
Salort. The point processes are the microscopic models, the PDE being their mesomacroscopic counterpart. In this sense, it extends PDE approaches for IF models to
models that statistically fit true spike trains data. In the first section, we introduce
Pakdaman, Perthame and Salort PDE (PPS) via its heuristic informal and microscopic description, which is based on IF models. Then, in Section 2, we develop
the different point process models, quite informally, to draw the main heuristic
correspondences between both approaches. In particular, we introduce the conditional intensity of a point process and a fundamental construction, called Ogata’s
thinning 29 , which allows a microscopic understanding of the dynamics of a point
process. Thanks to Ogata’s thinning, in Section 3, we have been able to rigorously
derive a microscopic random weak version of (PPS) and to propose its expectation deterministic counterpart. An independent and identically distributed (i.i.d)
population version is also available. Several examples of applications are discussed
in Section 4. To facilitate reading, technical results and proofs are included in two
appendices. The present work is clearly just a first to link point processes and PDE:
there are much more open questions than answered ones and this is discussed in the
final conclusion. However, we think that this can be fundamental to acquire a deeper
understanding of spike train models, their advantages as well as their limitations.

1. Synaptic integration and (PPS) equation
Based on the intuition that every neuron in the network should behave in the same
way, Pakdaman, Perthame and Salort proposed in 31 a deterministic PDE denoted
(PPS) in the sequel. The origin of this PDE is the classical (IF) model. In this section
we describe the link between the (IF) microscopic model and the mesoscopic (PPS)

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model, the main aim being to show thereafter the relation between (PPS) model
and other natural microscopic models for spike trains: point processes.
1.1. Integrate-and-fire
Integrate-and-fire models describe the time evolution of the membrane potential,
V (t), by means of ordinary differential equations as follows
dtV
= −gL (V − VL ) + I(t),
(1.1)
dt
where Cm is the capacitance of the membrane, gL is the leak conductance and VL
is the leak reversal potential. If V (t) exceeds a certain threshold θ, the neuron fires
/ emits an action potential (spike) and V (t) is reset to VL . The synaptic current
I(t) takes into account the fact that other presynaptic neurons fire and excite the
neuron of interest, whose potential is given by V (t).
As stated in 31 , the origin of (PPS) equation comes from 35 , where the explicit
solution of a classical IF model as (1.1) has been discussed. To be more precise the
membrane voltage of one neuron at time t is described by:
Z t
V (t) = Vr + (VL − Vr )e−(t−T )/τm +
h(t − u)Ninput (du),
(1.2)
Cm

T

where Vr is the resting potential satisfying VL < Vr < θ, T is the last spike emitted
by the considered neuron, τm is the time constant of the system (normally τm =
gL /Cm ), h is the excitatory post synaptic potential (EPSP) and Ninput is the sum
of Dirac masses at each spike of the presynaptic neurons. Since after firing, V (t)
is reset to VL < Vr , there is a refractory period when the neuron is less excitable
than at rest. The constant time τm indicatesR whether the next spike can occur
t
more or less rapidly. The other main quantity, T h(t − u)Ninput (du), is the synaptic
integration term.
In 35 , they consider a whole random network of such IF neurons and look at the
behavior of this model, where the only randomness is in the network. In many other
studies 7,8,9,11,26,42,30 IF models as (1.1) are considered to finally obtain other systems of partial differential equations (different to (PPS)) describing neural networks
behavior. In these studies, each presynaptic neuron is assumed to fire as an independent Poisson process and via a diffusion approximation, the synaptic current is then
approximated by a continuous in time stochastic process of Ornstein-Uhlenbeck.
1.2. The (PPS) equation
The deterministic PDE proposed by Pakdaman, Perthame and Salort, whose origin
is also the microscopic
IF model (1.2), is the following:
(
∂n(s,t)
+ ∂n(s,t)
+ p (s, X (t)) n (s, t) = 0
∂t
∂s
(PPS)
R +∞
m (t) := n (0, t) = 0 p (s, X (t)) n (s, t) ds.
In this equation, n(s, t) represents a probability density of neurons at time t having
discharged at time t − s. Therefore, s represents the time elapsed since the last

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discharge. The fact that the equation is an elapsed time structured equation is
natural, because the IF model (1.2) clearly only depends on the time since the last
spike. More informally, the variable s represents the ”age” of the neuron.
The first equation of the system (PPS) represents a pure transport process and
means that as time goes by, neurons of age s and past given by X(t) are either
aging linearly or reset to age 0 with rate p (s, X (t)).
The second equation of (PPS) describes the fact that when neurons spike, the
age (the elapsed time) returns to 0. Therefore, n(0, t) depicts the density of neurons
undergoing a discharge at time t and it is denoted by m(t). As a consequence of
this boundary condition, for n at s = 0, the following conservation law is obtained:
Z ∞
Z ∞
n (s, t) ds =
n (s, 0) ds
0

0

This means that if n (·, 0) is a probabilistic density then n (·, t) can be interpreted
as a density at each time t. Denoting by dt the Lebesgue measure and since m(t) is
the density of firing neurons at time t in (PPS), m(t)dt can also be interpreted as
the limit of Ninput (dt) in (1.2) when the population of neurons becomes continuous.
The system (PPS) is nonlinear since the rate p (s, X(t)) depends on n(0, t) by
means of the quantity X(t):
Zt

Zt
h(u)m(t − u)du =

X(t) =
0

h(u)n(0, t − u)du.

(1.3)

0

The quantity X(t) represents the interactions between neurons. It ”takes into account the averaged propagation time for the ionic pulse in this network” 31 . More
precisely with respect to the IF models (1.2), this is the synaptic integration term,
once the population becomes continuous. The only difference is that in (1.2) the
memory is cancelled once the last spike has occurred and this is not the case here.
However informally, both quantities have the same interpretation. Note nevertheless, that in 31 , the function h can be much more general than the h of the IF models
which clearly corresponds to EPSP. From now on and in the rest of the paper, h is
just a general non negative function without forcing the connection with EPSP.
The larger p (s, X(t)) the more likely neurons of age s and past X(t) fire. Most of
the time (but it is not a requisite), p is assumed to be less than 1 and is interpreted
as the probability that neurons of age s fire. However, as shown in Section 3 and
as interpreted in many population structured equation 14,19,34 , p(s, X(t)) is closer
to a hazard rate, i.e. a positive quantity such that p (s, X(t)) dt is informally the
probability to fire given that the neuron has not fired yet. In particular, it could be
not bounded by 1 and does not need to integrate to 1. A toy example is obtained
if p (s, X(t)) = λ > 0, where a steady state solution is n(s, t) = λe−λs 1s≥0 : this is
the density of an exponential variable with parameter λ.
However, based on the interpretation of p (s, X(t)) as a probability bounded
by 1, one of the main model that Pakdaman, Perthame and Salort consider is
p (s, X(t)) = 1s≥σ(X(t)) . This again can be easily interpreted by looking at (1.2).

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Indeed, since in the IF models the spike happens when the threshold θ is reached,
one can consider that p (s, X(t)) should be equal to 1 whenever
V (t) = Vr + (VL − Vr )e−(t−T )/τm + X(t) ≥ θ,
and 0 otherwise. Since VL − Vr < 0, p (s, X(t)) = 1 is indeed equivalent to s = t − T
larger than some decreasing function of X(t). This has the double advantage to give
a formula for the refractory period (σ(X(t))) and to model excitatory systems: the
refractory period decreases when the whole firing rate increases via X(t) and this
makes the neurons fire even more. This is for this particular case that Pakdaman,
Perthame and Salort have shown existence of oscillatory behavior 32 .
Another important parameter in the (PPS)
model and introduced in 31 is J,
R
which can be seen with our formalism as h and which describes the network
connectivity or the strength of the interaction. In 31 it has been proved that, for
highly or weakly connected networks, (PPS) model exhibits relaxation to steady
state and periodic solutions have also been numerically observed for moderately
connected networks. The authors in 32 have quantified the regime where relaxation
to a stationary solution occurs in terms of J and described periodic solution for
intermediate values of J.
Recently, in 33 , the (PPS) model has been extended including a fragmentation
term, which describes the adaptation and fatigue of the neurons. In this sense, this
new term incorporates the past activity of the neurons. For this new model, in
the linear case there is exponential convergence to the steady states, while in the
weakly nonlinear case a total desynchronization in the network is proved. Moreover,
for greater nonlinearities, synchronization can again been numerically observed.
2. Point processes and conditional intensities as models for spike
trains
We first start by quickly reviewing the main basic concepts and notations of point
processes, in particular, conditional intensities and Ogata’s thinning 29 . We refer
the interested reader to 3 for exhaustiveness and to 6 for a much more condensed
version, with the main useful notions.
2.1. Counting processes and conditional intensities
We focus on locally finite point processes on R, equipped with the borelians B(R).
Definition 2.1 (Locally finite point process). A locally finite point process N
on R is a random set of points such that it has almost surely (a.s.) a finite number
of points in finite intervals. Therefore, associated to N there is an ordered sequence
of extended real valued random times (Tz )z∈Z : · · · ≤ T−1 ≤ T0 ≤ 0 < T1 ≤ · · · .
For a measurable set A, NA denotes the number of points of N in A. This is a
random variable with values in N ∪ {∞}.
Definition 2.2 (Counting process associated to a point process). The process on R+ defined by t 7→ Nt := N(0,t] is called the counting process associated to
the point process N .

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The natural and the predictable filtrations are fundamental for the present work.
Definition 2.3 (Natural
 filtration of a point process). The natural filtration
of N is the family FtN t∈R of σ-algebras defined by FtN = σ (N ∩ (−∞, t]).
Definition 2.4 (Predictable filtration of a point
 process). The NpreN
defined by Ft− =
dictable filtration of N is the family of σ-algebra Ft−
t∈R
σ (N ∩ (−∞, t)).
The intuition behind this concept is that FtN contains all the information given
by the point process at time t. In particular, it contains the information whether
N
only contains the information given
t is a point of the process or not while Ft−
by the point process strictly before t. Therefore, it does not contain (in general)
N
represents (the
the information whether t is a point or not. In this sense, Ft−
information contained in) the past.
Under some rather classical conditions 3 , which are always assumed to be satN
), which is
isfied here, one can associate to (Nt )t≥0 a stochastic intensity λ(t, Ft−
N
a non negative random quantity. The notation λ(t, Ft− ) for the intensity refers
to the Rpredictable version of the intensity associated to the natural filtration and
t
N
N
(Nt − 0 λ(u, Fu−
)du)t≥0 forms a local martingale 3 . Informally, λ(t, Ft−
)dt represents the probability to have a new point in interval [t, t + dt) given the past. Note
N
) should not be understood as a function, in the same way as density
that λ(t, Ft−
is for random variables. It is a ”recipe” explaining how the probability to find a
new point at time t depends on the past configuration: since the past configuration
depends on its own past, this is closer to a recursive formula. In this respect, the
intensity should obviously depend on N ∩(−∞, t) and not on N ∩(−∞, t] to predict
the occurrence at time t, since we cannot know whether t is already a point or not.
The distribution of the point process N on R is completely characterized by the
N
) on R+ and the distribution of N− = N ∩ R− ,
knowledge of the intensity λ(t, Ft−
which is denoted by P0 in the sequel. The information about P0 is necessary since
each point of N may depend on the occurrence of all the previous points: if for all
N
t > 0, one knows the ”recipe” λ(t, Ft−
) that gives the probability of a new point at
time t given the past configuration, one still needs to know the distribution of N−
to obtain the whole process.
Two main assumptions
are used depending on the type of results we seek:

1
RT
,a.s.
N
ALλ,loc
for any T ≥ 0, 0 λ(t, Ft−
)dt is finite a.s.
 1

hR
i
T
,exp
N
ALλ,loc
for any T ≥ 0, E 0 λ(t, Ft−
)dt is finite.
 1

 1

 1

Clearly ALloc,exp implies ALloc,a.s. . Note that ALloc,a.s. implies non-explosion in
finite time for the counting processes (Nt ).
Definition 2.5 (Point measure associated to a point process). The point
P
measure associated to N is denoted by N (dt) and defined by N (dt) = i∈Z δTi (dt),
where δu is the Dirac mass in u.
By analogy with (PPS), and since points of point processes correspond to spikes

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(or times of discharge) for the considered neuron in spike train analysis, N (dt) is the
microscopic equivalent of the distribution of discharging neurons m(t)dt. Following
this analogy, and since TNt is the last point less or equal to t for every t ≥ 0, the
age St at time t is defined by St = t − TNt . In particular, if t is a point of N , then
St = 0. Note that St is FtN measurable for every t ≥ 0 and therefore, S0 = −T0 is
F0N measurable. To define an age at time t = 0, one assumes that
(AT0 ) There exists a first point before 0 for the process N− , i.e. −∞ < T0 .
As we have remarked before, conditional intensity should depend on N ∩ (−∞, t).
Therefore, it cannot be function of St , since St informs us if t is a point or not.
N
That is the main reason for considering this Ft−
measurable variable
St− = t − TNt− ,

(2.1)

where TNt− is the last point strictly before t (see Figure 1). Note also that knowing
(St− )t≥0 or (Nt )t≥0 is completely equivalent given F0N .
The last and most crucial equivalence between (PPS) and the present point
N
process set-up, consists in noting that the quantities p(s, X(t)) and λ(t, Ft−
) have
informally the same meaning: they both represent a firing rate, i.e. both give the
rate of discharge as a function of the past. This dependence is made more explicit
N
).
in p(s, X(t)) than in λ(t, Ft−
2.2. Examples
Let us review the basic point processes models of spike trains and see what kind of
analogy is likely to exist between both models ((PPS) equation and point processes).
These informal analogies are possibly exact mathematical results (see Section 4).
N
Homogeneous Poisson process This is the simplest case where λ(t, Ft−
) = λ,
with λ a fixed positive constant representing the firing rate. There is no dependence
in time t (it is homogeneous) and no dependence with respect to the past. This
case should be equivalent to p(s, X(t)) = λ in (PPS). This can be made even more
explicit. Indeed in the case where the Poisson process exists on the whole real line
(stationary case), it is easy to see that


P (St− > s) = P N[t−s,t) = 0 = exp(−λs),
meaning that the age St− obeys an exponential distribution with parameter λ, i.e.
the steady state of the toy example developed for (PPS) when p(s, X(t)) = λ.
Inhomogeneous Poisson process To model non stationarity, one can use
N
λ(t, Ft−
) = λ(t), which only depends on time. This case should be equivalent to the
replacement of p(s, X(t)) in (PPS) by λ(t).
Renewal process This model is very useful to take refractory period into account.
It corresponds to the case where the ISIs (delays between spikes) are independent
and identically distributed (i.i.d.) with a certain given density ν on R+ . The asso-

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ciated hazard rate is
ν(s)
f (s) = R +∞
,
ν(x)dx
s
R +∞
when s ν(x)dx > 0. Roughly speaking, f (s)ds is the probability that a neuron
spikes with age s given that its age is larger than s. In this case, considering the set
of spikes as the point process N , it is easy to show (see the Appendix B.1) that its
N
corresponding intensity is λ(t, Ft−
) = f (St− ) which only depends on the age. One
can also show quite easily that the process (St− )t>0 , which is equal to (St )t>0 almost
everywhere (a.e.), is a Markovian process in time. This renewal setting should be
equivalent in the (PPS) framework to p(s, X(t)) = f (s).
Note that many people consider IF models (1.2) with Poissonian inputs with or
without additive white noise. In both cases, the system erases all memory after each
spike and therefore the ISIs are i.i.d. Therefore as long as we are only interested by
the spike trains and their point process models, those IF models are just a particular
case of renewal process 8,10,17,35 .
Wold process and more general structures Let A1t be the delay (ISI) between
the last point and the occurrence just before (see also Figure 1), A1t = TNt− −TNt− −1 .
N
) = f (St− , A1t ). This model
A Wold process 24,16 is then characterized by λ(t, Ft−
has been matched to several real data thanks to goodness-of-fit tests 38 and is
therefore one of our main example with the next discussed Hawkes process case.
One can show in this case that the successive ISI’s form a Markov chain of order 1
and that the continuous time process (St− , A1t ) is also Markovian.
This case should be equivalent to the replacement of p(s, X(t)) in (PPS) by
f (s, a1 ), with a1 denoting the delay between the two previous spikes. Naturally in
this case, one should expect a PDE of higher dimension with third variable a1 .
More generally, one could define
Akt = TNt− −(k−1) − TNt− −k ,

(2.2)

N
and point processes with intensity λ(t, Ft−
) = f (St− , A1t , ..., Akt ). Those processes
satisfy more generally that their ISI’s form a Markov chain of order k and that the
continuous time process (St− , A1t , ..., Akt ) is also Markovian (see the Appendix B.2).

Remark 2.1. The dynamics of the successive ages is pretty simple. On the one
hand, the dynamics of the vector of the successive ages (St− , A1t , ..., Akt )t>0 is deterministic between two jumping times. The first coordinate increases with rate 1.
On the other hand, the dynamics at any jumping time T is given by the following
shift:


 the age process goes to 0, i.e. ST = 0,
(2.3)
the first delay becomes the age, i.e. A1T + = ST − ,


i−1
i
the other delays are shifted, i.e. AT + = AT for all i ≤ k.

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Hawkes processes The most classical setting is the linear (univariate) Hawkes
process, which corresponds to
N
λ(t, Ft−
)

t−

Z

h(t − u)N (du),

=µ+
−∞

where the positive parameter µ is called the spontaneous rate and the non negative
function h, with support
in R+ , is called the interaction function, which is generally
R
assumed to satisfy R+ h < 1 to guarantee the existence of a stationary version 16 .
This model has also been matched to several real neuronal data thanks to goodnessof-fit tests 40 . Since it can mimic synaptic integration, as explained below, this
represents the main example of the present work.
In the case where T0 tends to −∞, this is equivalent to say that there is no point
on the negative half-line and in this case, one can rewrite
N
λ(t, Ft−
)

Z

t−

h(t − u)N (du).

=µ+
0

R t−
By analogy between N (dt) and m(t)dt, one sees that 0 h(t − u)N (du) is indeed
the analogous of X(t) the synaptic integration in (1.3). So one could expect that
the PDE analogue is given by p(s, X(t)) = µ + X(t). In Section 4, we show that
this does not hold stricto sensu, whereas the other analogues work well.
Note that this model shares also some link with IF models. Indeed, the formula
for the intensity is close to the formula for the voltage (1.2), with the same flavor for
the synaptic integration term. The main difference comes from the fact that when
the voltage reaches a certain threshold, it fires deterministically for the IF model,
whereas the higher the intensity, the more likely is the spike for the Hawkes model,
but without certainty. In this sense Hawkes models seem closer to (PPS) since as we
discussed before, the term p(s, X(t)) is closer to a hazard rate and never imposes
deterministically the presence of a spike.
To model inhibition (see 41 for instance), one can use functions h that may take
R t−
N
negative values and in this case λ(t, Ft−
) = µ + −∞ h(t − u)N (du) , which
+

N
should correspond to p(s, X(t)) = (µ + X(t))+ . Another possibility is λ(t, Ft−
)=


R t−
exp µ + −∞ h(t − u)N (du) , which is inspired by the generalized linear model as

used by 36 and which should correspond to p(s, X(t)) = exp (µ + X(t)).
Note finally that Hawkes models in Neuroscience (and their variant) are usually
multivariate meaning that they model interaction between spike trains thanks to
interaction functions between point processes, each process representing a neuron.
To keep the present analogy as simple as possible, we do not deal with those multivariate models in the present article. Some open questions in this direction are
presented in conclusion.

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11

2.3. Ogata’s thinning algorithm
N
To turn the analogy between p(s, X(t)) and λ(t, Ft−
) into a rigorous result on the
PDE level, we need to understand the intrinsic dynamics of the point process. This
dynamics is often not explicitly described in the literature (see e.g. the reference
book by Brémaud 3 ) because martingale theory provides a nice mathematical setting in which one can perform all the computations. However, when one wants to
simulate point processes based on the knowledge of their intensity, there is indeed
a dynamics that is required to obtain a practical algorithm. This method has been
described at first by Lewis in the Poisson setting 25 and generalized by Ogata in
29
. If there is a sketch of proof in 29 , we have been unable to find any complete
mathematical proof of this construction in the literature and we propose a full and
mathematically complete version of this proof with minimal assumptions in the
Appendix B.4. Let us just informally describe here, how this construction works.
The principle consists in assuming that is given an external homogeneous Poisson
process Π of intensity 1 in R2+ and with associated point measure Π (dt, dx) =
P
(T,V )∈Π δ(T,V ) (dt, dx). This means in particular that

E [Π(dt, dx)] = dt dx.

(2.4)

Once a realisation of N− fixed, which implies that F0N is known and which can be
seen as an initial condition for the dynamics, the construction of the process N on
R+ only depends on Π.
N
) in the sense of the ”recipe”
More precisely, if we know the intensity λ(t, Ft−
that explicitly depends on t and N ∩(−∞, t), then once a realisation of Π and of N−
N
) for t ∈ R+
is fixed, the dynamics to build a point process N with intensity λ(t, Ft−
is purely deterministic. It consists (see also Figure 1) in successively projecting on
N
the abscissa axis the points that are below the graph of λ(t, Ft−
). Note that a point
N
projection may change the shape of λ(t, Ft− ), just after the projection. Therefore the
N
graph of λ(t, Ft−
) evolves thanks to the realization of Π. For a more mathematical
description, see Theorem B.11 in the Appendix B.4.
 Note in particular that the
construction ends on any finite interval [0, T ] a.s. if A1,a.s
λ,loc holds.
Then the point process N , result of Ogata’s thinning, is given by the union of
N
N− on R− and the projected points on R+ . It admits the desired intensity λ(t, Ft−
)
on R+ . Moreover, the point measure can be represented by
1t>0 N (dt) =

X
(T,X)∈Π /
X≤λ(T,FTN− )

N
Z λ(t,Ft−
)

δT (dt) =

!
Π (dt, dx) .

(2.5)

x=0

NB: The last equality comes from the following convention. If δ(c,d) is a Dirac mass
Rb
in (c, d) ∈ R2+ , then x=a δ(c,d) (dt, dx), as a distribution in t, is δc (dt) if d ∈ [a, b]
and 0 otherwise.

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Fig. 1. Example of Ogata’s thinning algorithm on a linear Hawkes process with interaction function
h(u) = e−u and no point before 0 (i.e. N− = ∅). The crosses represent a realization of Π, Poisson
N ),
process of intensity 1 on R2+ . The blue piecewise continuous line represents the intensity λ(t, Ft−
which starts in 0 with value µ and then jumps each time a point of Π is present underneath it.
N )) is given by the blue circles. Age S
The resulting Hawkes process (with intensity λ(t, Ft−
t− at
1
time t and the quantity At are also represented.

3. From point processes to PDE
Let us now present our main results. Informally, we want to describe the evolution
of the distribution in s of the age St according to the time t. Note that at fixed time
t, St− = St a.s. and therefore it is the same as the distribution of St− . We prefer
to study St− since its predictability, i.e. its dependence in N ∩ (−∞, t), makes all
definitions proper from a microscopic/random point of view. Microscopically, the
interest lies in the evolution of δSt− (ds) as a random measure. But it should also be
seen as a distribution in time, for equations like (PPS) to make sense. Therefore,
we need to go from a distribution only in s to a distribution in both s and t. Then
one can either focus on the microscopic level, where the realisation of Π in Ogata’s
thinning construction is fixed or focus on the expectation of such a distribution.
3.1. A clean setting for bivariate distributions in age and time
In order to obtain, from a point process, (PPS) system we need to define bivariate
distributions in s and t and marginals (at least in s), in such a way that weak solutions of (PPS) are correctly defined. Since we want to possibly consider more than
two variables for generalized Wold processes, we consider the following definitions.
In the following, < ϕ, ν > denotes the integral of the integrable function ϕ with
respect to the measure ν.
Let k ∈ N. For every bounded measurable function ϕ of (t, s, a1 , ..., ak ) ∈ Rk+2
+ ,
one can define
(1)

ϕt (s, a1 , ..., ak ) = ϕ(t, s, a1 , ..., ak )

and ϕ(2)
s (t, a1 , ..., ak ) = ϕ(t, s, a1 , ..., ak ).

Let us now define two sets of regularities for ϕ.

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Mc,b (Rk+2
+ )

13

The function ϕ belongs to Mc,b (Rk+2
+ ) if and only if
• ϕ is a measurable bounded function,
(1)
• there exists T > 0 such that for all t > T , ϕt = 0.

∞
The function ϕ belongs to Cc,b
(Rk+2
+ ) if and only if
•
ϕ
is
continuous,
uniformly
bounded,
∞
Cc,b
(Rk+2
+ )
• ϕ has uniformly bounded derivatives of every order,
(1)
• there exists T > 0 such that for all t > T , ϕt = 0.

Let (ν1t )t≥0 be a (measurable w.r.t. t) family of positive measures on Rk+1
+ , and
k+1
s
(ν2 )s≥0 be a (measurable w.r.t. s) family of positive measures R+ . Those families
satisfy the Fubini property if
R (1) t
R (2)
(PF ubini ) for any ϕ ∈ Mc,b (Rk+2
hϕt , ν1 idt = hϕs , ν2s ids.
+ ),
k+2
In this case, one can define ν, measure on Rk+2
+ , by the unique measure on R+
such that for any test function ϕ in Mc,b (Rk+2
+ ),
Z
Z
(1)
s
< ϕ, ν >= hϕt , ν1t idt = hϕ(2)
s , ν2 ids.

To simplify notations, for any such measure ν(t, ds, da1 , ..., dak ), we define
ν(t, ds, da1 , ..., dak ) = ν1t (ds, da1 , ..., dak ),

ν(dt, s, da1 , ..., dak ) = ν2s (dt, da1 , ..., dak ).

In the sequel, we need in particular a measure on R2+ , ηx , defined for any real x
by its marginals that satisfy (PF ubini ) as follows
∀ t, s ≥ 0,

ηx (t, ds) = δt−x (ds)1t−x>0

and ηx (dt, s) = δs+x (dt)1s≥0 . (3.1)

It represents a Dirac mass ”travelling” on the positive diagonal originated in (x, 0).
3.2. The microscopic construction of a random PDE
For a fixed realization of Π, we therefore want to define a random distribution
U (dt, ds) in terms of its marginals, thanks to (PF ubini ), such that, U (t, ds) represents the distribution at time t > 0 of the age St− , i.e.
∀ t > 0,

U (t, ds) = δSt− (ds)

(3.2)

and satisfies similar equations as (PPS). This is done in the following proposition.

N
Proposition 3.1. Let Π, F0N and an intensity λ(t, Ft−
) t>0 be given as in Section
 1

,a.s.
2.3 and satisfying (AT0 ) and ALλ,loc
. On the event Ω of probability 1, where
Ogata’s thinning is well defined, let N be the point process on R that is constructed
thanks to Ogata’s thinning with associated predictable age process (St− )t>0 and
whose points are denoted (Ti )i∈Z . Let the (random) measure U and its corresponding
marginals be defined by
U (dt, ds) =

+∞
X
i=0

ηTi (dt, ds) 10≤t≤Ti+1 .

(3.3)

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Then, on Ω, U satisfies (PF ubini ) and U (t, ds) = δSt− (ds). Moreover, on Ω, U is a
solution in the weak sense of the following system
!
N
Z λ(t,Ft−
)
∂
∂
U (dt, ds) +
U (dt, ds) +
Π (dt, dx) U (t, ds) = 0,
∂t
∂s
x=0
!
N
Z
Z λ(t,Ft−
)
U (dt, 0) =
Π (dt, dx) U (t, ds) + δ0 (dt)1T0 =0 ,
s∈R+

(3.4)

(3.5)

x=0

U (0, ds) = δ−T0 (ds)1T0 <0 = U in (ds)1s>0 ,

(3.6)

∞
where U in (ds) = δ−T0 (ds). The weak sense means that for any ϕ ∈ Cc,b
(R2+ ),


Z
∂
∂
ϕ (t, s) +
ϕ (t, s) U (dt, ds) +
∂t
∂s
R+ ×R+
!
N
Z
Z λ(t,Ft−
)
[ϕ (t, 0) − ϕ (t, s)]
Π (dt, dx) U (t, ds) + ϕ(0, −T0 ) = 0. (3.7)
R+ ×R+

x=0

The proof of Proposition 3.1 is included in Appendix A.1. Note also that thanks to
the Fubini property, the boundary condition (3.5) is satisfied also in a strong sense.
System (3.4)–(3.6) is a random microscopic version of (PPS) if T0 < 0, where
n(s, t) the density of the age at time t is replaced by U (t, ·) = δSt− , the Dirac mass
in the age at time t. The assumption T0 < 0 is satisfied a.s. if T0 has a density, but
this may not be the case for instance if the experimental device gives an impulse
at time zero (e.g. 38 studied Peristimulus time histograms (PSTH), where the spike
trains are locked on a stimulus given at time 0).
This result may seem rather poor from a PDE point of view. However, since this
equation is satisfied at a microscopic level, we are able to define correctly all the
important quantities at a macroscopic level. Indeed, the analogy between p(s, X(t))
N
and λ(t, Ft−
) is actually on the random microscopic scale a replacement of p(s, X(t))
N
R λ(t,Ft−
)
Π (dt,dx), whose expectancy given the past is, heuristically speaking,
by x=0
N
equal to λ t, Ft−
because the mean behaviour of Π is given by the Lebesgue
measure (see (2.4)). Thus, the main question at this stage is : can we make this
argument valid by taking the expectation of U ? This is addressed in the next section.
The property (PF ubini ) and the quantities ηTi mainly allows to define U (dt, 0)
as well as U (t, ds). As expected, with this definition, (3.2) holds as well as
U (dt, 0) = 1t≥0 N (dt),

(3.8)

i.e. the spiking measure (the measure in time with age 0) is the point measure.
Note also that the initial condition is given by F0N , since F0N fixes in particular
the value of T0 and (AT0 ) is required to give sense to the age at time 0. To understand
the initial condition, remark that if T0 = 0, then U (0, ·) = 0 6= limt→0+ U (t, ·) = δ0
by definitions of ηTi but that if T0 < 0, U (0, ·) = limt→0+ U (t, ·) = δ−T0 .

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R∞
The conservativeness (i.e. for all t ≥ 0, 0 U (t, ds) = 1) is obtained by using (a
sequence of test functions converging to) ϕ = 1t≤T .
Proposition 3.1 shows that the (random) measure U , defined by (3.3), in terms
of a given point process N , is a weak solution of System (3.4)-(3.6). The study of
the well-posedness of this system could be addressed following, for instance, the
ideas given in 12 . In this case U should be the unique solution of system (3.4)–(3.6).
As last comment about Proposition 3.1, we analyse the particular case of the
linear Hawkes process, in the following remark.
R t−
N
) = µ + −∞ h(t − z)N (dz).
Remark 3.1. In the linear Hawkes process, λ(t, Ft−
Thanks to (3.8) one decomposes the intensity into a term given Rby the initial condit−
N
) = µ+F0 (t)+ 0 h(t−z)U (dz, 0),
tion plus a term given by the measure U , λ(t, Ft−
R0
where F0 (t) = −∞ h(t − z)N− (dz) is (F0N )-measurable and considered as an initial
N
) is
condition. Hence, (3.4)–(3.6) becomes a closed system in the sense that λ(t, Ft−
now an explicit function of the solution of the system. This is not true in general.
3.3. The PDE satisfied in expectation
In this section, we want to find the system satisfied by the expectation of the
random measure U . First, we need to give a proper definition of such an object. The
construction is based on the construction of U and is summarized in the following
proposition. (The proofs of all the results of this subsection are in Appendix A.1).

N
Proposition 3.2. Let Π, F0N and an intensity λ(t, Ft−
) t>0 be given as in Section
 1

,exp
2.3 and satisfying (AT0 ) and ALλ,loc
. Let N be the process resulting of Ogata’s
thinning and let U be the random measure defined by (3.3). Let E denote the expectation with respect to Π and F0N .
R

2
RThen for any test function ϕ in Mc,b (R+ ), E ϕ(t, s)U (t, ds) and
E ϕ(t, s)U (dt, s) are finite and one can define u(t, ds) and u(dt, s) by

Z

Z


ϕ(t, s)u(t, ds) = E
ϕ(t, s)U (t, ds) ,
∀ t ≥ 0,
Z

R


ϕ(t, s)u(dt, s) = E
ϕ(t, s)U (dt, s) .
∀ s ≥ 0,
Moreover, u(t, ds) and u(dt, s) satisfy (PF ubini ) and one can define u(dt, ds) =
u(t, ds)dt = u(dt, s)ds on R2+ , such that for any test function ϕ in Mc,b (R2+ ),
Z

Z
ϕ(t, s)u(dt, ds) = E
ϕ(t, s)U (dt, ds) ,
quantity which is finite.
R

R
In particular, since ϕ(t, s)u(t, ds) = E ϕ(t, s)U (t, ds) = E [ϕ(t, St− )], u(t, ·)
is therefore the distribution of St− , the (predictable version of the) age at time t.
Now let us show that as expected, u satisfies a system similar to (PPS).

N
Theorem 3.3. Let Π, F0N and an intensity λ(t, Ft−
) t>0 be given as in Section

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2.3 and satisfying (AT0 ) and




L1 ,exp
Aλ,loc
. If N is the process resulting of Ogata’s

thinning, (St− )t>0 its associated predictable age process, U its associated random
measure, defined by (3.3), and u its associated mean measure, defined in Proposition 3.2, then, there exists a bivariate measurable function ρλ,P0 satisfying

Z TZ

 ∀ T ≥ 0,
ρλ,P0 (t, s)u(dt, ds) < ∞,
(3.9)
0
s





N
u(dt, ds)- a.e
ρλ,P0 (t, s) = E λ t, Ft− St− = s
and such that u is solution in the weak sense of the following system
∂
∂
u (dt, ds) +
u (dt, ds) + ρλ,P0 (t, s)u (dt, ds) = 0,
∂t
∂s
Z

ρλ,P0 (t, s)u (t, ds) dt + δ0 (dt)uin ({0}),

u (dt, 0) =

(3.10)
(3.11)

s∈R+

u (0, ds) = uin (ds)1s>0 ,

(3.12)

∞
where uin is the law of −T0 . The weak sense means here that for any ϕ ∈ Cc,b
(R2+ ),


Z
∂
∂
+
ϕ (t, s) u (dt, ds) +
∂t ∂s
R+ ×R+
Z
Z
[ϕ(t, 0) − ϕ(t, s)]ρλ,P0 (t, s)u(dt, ds) +
ϕ(0, s)uin (ds) = 0, (3.13)
R+ ×R+

R+

Comparing this system to (PPS), one first sees that n(·, t), the density of the
age at time t, is replaced by the mean measure u(t, ·). If uin ∈ L1 (R+ ) we have
uin ({0}) = 0 so we get an equation which is exactly of renewal type, as (PPS).
In the general case where uin is only a probability measure, the difference with
(PPS) lies in the term δ0 (dt)uin ({0}) in the boundary condition for s = 0 and in
the term 1s>0 in the initial condition for t = 0. Both these extra terms are linked
to the possibility for the initial measure uin to charge zero. This possibility is not
considered in 31 - else, a similar extra term would be needed in the setting of 31 as
well. As said above in the comment of Proposition 3.1, we want to keep this term
here since it models the case where there is a specific stimulus at time zero 38 .
In general and without more assumptions on λ, it is not clear that u is not only
a measure satisfying (PF ubini ) but also absolutely continuous wrt to dt ds and that
the equations can be satisfied in a strong sense.
Concerning p(s, X(t)), which has always been thought of as the equivalent of
N
N
λ(t, Ft−
), it is not replaced by λ(t, Ft−
), which would
 have no meaning in general
N
since this is a random quantity, nor
by
E
λ(t,
F
)
t− which would have been a first

N
possible guess; it is replaced by E λ(t, Ft− )|St− = s . Indeed intuitively, since
"Z
#
N
λ(t,Ft−
)

N
N
= λ t, Ft−
dt,
E
Π (dt, dx) Ft−
x=0

the corresponding weak term can be interpreted as, for any test function ϕ,

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N
Z λ(t,Ft−
)

"Z
E

ϕ (t, s)

!

#

Z

Π (dt, dx) U (t, ds) = E

ϕ (t, s) λ

N
t, Ft−



17


δSt− (ds)dt

x=0

Z
=
Zt
=



N
E ϕ (t, St− ) λ t, Ft−
dt




N
E ϕ (t, St− ) E λ t, Ft−
|St− dt,

t

R

which is exactly ϕ(t, s)ρλ,P0 (t, s)u(dt, ds).
This conditional expectation makes dependencies particularly complex, but this
also enables to derive equations even in non-Markovian setting (as Hawkes processes
for instance, see Section 4). More explicitly, ρλ,P0 (t, s) is a function of the time t,
of the age s, but it also depends on λ, the shape of the intensity of the underlying
process and on the distribution of the initial condition N− , that is P0 . As explained
in Section 2, it is both the knowledge of P0 and λ that characterizes the distribution
of the process and in general the conditional expectation cannot be reduced to
something depending on less than that. In Section 4, we discuss several examples
of point processes where one can (or cannot) reduce the dependence.
Note that here again, we can prove that the equation is conservative by taking
(a sequence of functions converging to) ϕ = 1t≤T as a test function.
A direct corollary of Theorem 3.3 can be deduced thanks to the law of large
numbers. This can be seen as the interpretation of (PPS) equation at a macroscopic
level, when the population of neurons is i.i.d..
∞
Corollary 3.4. Let N i i=1 be some i.i.d. point processes with intensity given by
 1

,exp
Ni
λ(t, Ft−
) on (0, +∞) satisfying ALλ,loc
and associated predictable age processes
i
(St−
)t>0 . Suppose furthermore that the distribution of N 1 on (−∞, 0] is given by
1
P0 which is such that P0 (N−
= ∅) = 0.
Then there exists a measure u satisfying (PF ubini ), weak solution of Equations (3.10) and (3.11), with ρλ,P0 defined by
h 

i
N1
1
ρλ,P0 (t, s) = E λ t, Ft−
|St−
= s , u(dt, ds) − a.e.
∞
and with uin distribution of the age at time 0, such that for any ϕ ∈ Cc,b
(R2+ )
!
Z
Z
n
1X
a.s.
δ i (ds) −−−−→ ϕ(t, s)u(t, ds),
(3.14)
∀ t > 0,
ϕ(t, s)
n→∞
n i=1 St

In particular, informally, the fraction of neurons at time t with age in [s, s + ds)
in this i.i.d. population of neurons indeed tends to u(t, ds).
4. Application to the various examples
Let us now apply these results to the examples presented in Section 2.2.

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4.1. When the intensity only depends on time and age

N
= f (t, St− ) (homogeneous and inhomogeneous Poisson processes and
If λ t, Ft−
renewal processes are particular
examples) then the intuition giving that p(s, X(t))

N
is analogous to λ t, F
works.
Let us assume that f (t, s) ∈ L∞ (R2+ ). We have
t−

N
E λ t, Ft−
|St− = s = f (t, s). Under this assumption, we may apply Theorem 3.3,
so that we know that the mean measure u associated to the random process is a solution of System (3.10)–(3.12). Therefore the mean measure u satisfies a completely
explicit PDE of the type (PPS) with ρλ,P0 (t, s) = f (t, s) replacing p(s, X(t)). In
particular, in this case ρλ,P0 (t, s) does not depend on the initial condition. As already underlined,
in general, the distribution of the process is characterized by

N
= f (t, St− ) and by the distribution of N− . Therefore, in this special
λ t, Ft−
case, this dependence is actually reduced
to the function f and the distribution of

−T0 . Since f (·, ·) ∈ L∞ [0, T ] × R+ , assuming also uin ∈ L1 (R+ ), it is well-known

that there exists a unique solution u such that (t 7→ u(t, ·)) ∈ C [0, T ], L1 (R+ ) ,
see for instance 34 Section 3.3. p.60. Note that following 12 uniqueness for measure solutions may also be established, hence the mean measure u associated to
the random process
is the unique solution of System (3.10)–(3.12), and it is in

1
C [0, T ], L (R+ ) : the PDE formulation, together with existence and uniqueness,
has provided a regularity result on u which is obtained under weaker assumptions
than through Fokker-Planck / Kolmogorov equations. This is another possible application field of our results: using the PDE formulation to gain regularity. Let us
now develop the Fokker-Planck / Kolmogorov approach for renewal processes.

N
Renewal processes The renewal process, i.e. when λ t, Ft−
= f (St− ), with f a
continuous function on R+ , has particular properties. As noted in Section 2.2, the
renewal age process (St− )t>0 is an homogeneous Markovian process. It is known
for a long time that it is easy to derive PDE on the corresponding density through
Fokker-Planck / Kolmogorov equations, once the variable of interest (here the age)
is Markovian (see for instance 1 ). Here we briefly follow this line to see what kind of
PDE can be derived through the Markovian properties and to compare the equation
with the (PPS) type system derived in Theorem 3.3.
Since f is continuous, the infinitesimal generatorb of (St )t>0 is given by
(Gφ)(x) = φ0 (x) + f (x) (φ(0) − φ(x)) ,

(4.1)

for all φ ∈ C 1 (R+ ) (see 2 ). Note that, since for every t > 0 St− = St a.s., the process
(St− )t>0 is also Markovian with the same infinitesimal generator.
b The

infinitesimal generator of an homogeneous Markov process (Zt )t≥0 is the operator G which
is defined to act on every function φ : Rn → R in a suitable space D by
Gφ(x) = lim

t→0+

E [φ(Zt )|Z0 = x] − φ(x)
.
t

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19

Let us now define for all t > 0 and all φ ∈ C 1 (R+ ),
Z
Pt φ(x) = E [φ(St− )|S0 = x] = φ(s)ux (t, ds),
where x ∈ R+ and ux (t, ·) is the distribution of St− given that S0 = x. Note
that ux (t, ds) corresponds to the marginal in the sense of (PF ubini ) of ux given by
Theorem 3.3 with ρλ,P0 (t, s) = f (s) and initial condition δx , i.e. T0 = −x a.s.
In this homogeneous Markovian case, the forward Kolmogorov equation gives
∂
Pt = Pt G.
∂t
∞
Let ϕ ∈ Cc,b
(R2+ ) and let t > 0. This implies that

∂
∂
(Pt ϕ(t, s)) = Pt Gϕ(t, s) + Pt ϕ(t, s)
∂t
∂t


∂
∂
= Pt
ϕ(t, s) + f (s) (ϕ(t, 0) − ϕ(t, s)) + ϕ(t, s) .
∂s
∂t
Since ϕ is compactly supported in time, an integration with respect to t yields


Z
Z
∂
∂
−P0 ϕ(0, s) = Pt
+
ϕ(t, s)dt + Pt f (s) (ϕ(t, 0) − ϕ(t, s)) dt,
∂t ∂s
or equivalently

Z
Z 
∂
∂
+
ϕ (t, s) ux (t, ds) dt − (ϕ(t, s) − ϕ(t, 0))f (s)ux (t, ds)dt,
− ϕ(0, x) =
∂t ∂s
(4.2)
in
in terms of ux . This is exactly Equation (3.13) with u = δx .
The result of Theorem 3.3 is stronger than the application of the forward Kolmogorov equation on homogeneous Markovian systems since the result of Theorem
3.3 never used the Markov assumption and can be applied to non Markovian processes (see Section 4.3). So the present work is a general set-up where one can deduce
PDE even from non Markovian microscopic random dynamics. Note also that only
boundedness assumptions and not continuity ones are necessary to directly obtain
(4.2) via Theorem 3.3: to obtain the classical Kolmogorov theorem, one would have
assumed f ∈ C 0 (R2+ ) rather than f ∈ L∞ (R2+ ).
4.2.

Generalized Wold process

N
= f (St− , A1t , ..., Akt ), with f being a non-negative funcIn the case where λ t, Ft−
tion, one can define in a similar way uk (t, s, a1 , . . . , ak ) which is informally the
distribution at time t of the processes with age s and past given by a1 , ...ak for
the last k ISI’s. We want to investigate this case not for its Markovian properties,
which are nevertheless presented in Proposition B.2 in the appendix for sake of
completeness, but because this is the first basic example where the initial condition
is indeed impacting ρλ,P0 in Theorem 3.3.

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To do so, the whole machinery
applied on u(dt, ds) is first extended in the next result

1
k
to uk dt, ds, da , . . . , da which represents the dynamics of the age and the last
k ISI’s. This could have been done in a very general way by an easy generalisation
of Theorem 3.3. However to avoid too cumbersome equations, we express it only
for generalized Wold processes to provide a clean setting to illustrate the impact
of the initial conditions on ρλ,P0 . Hence, we similarly define a random distribution
Uk (dt, ds, da1 , . . . , dak ) such that its evaluation at any given time t exists and is
Uk (t, ds, da1 , . . . , dak ) = δ(St− ,A1t ,...,Akt ) (ds, da1 , . . . , dak ).

(4.3)

The following result states the PDE satisfied by uk = E [Uk ].
Proposition 4.1. Let k be a positive integer and f be some non negative function on Rk+1
+ . Let N be a generalized Wold process with predictable age process
N
) = f (St− , A1t , ..., Akt ) sat(St− )t>0
points (Ti )i∈Z and intensity λ(t, Ft−
 , associated

1

,exp
isfying ALλ,loc
, where A1t , . . . , Akt are the successive ages defined by (2.2). Suppose

that P0 is such that P0 (T−k > −∞) = 1. Let Uk be defined by
Uk (dt, ds, da1 , . . . , dak ) =

+∞
X
i=0

ηTi (dt, ds)

k
Y
j=1

δAj (daj ) 10≤t≤Ti+1 ,

(4.4)

Ti

If N is the result of Ogata’s thinning on the Poisson process Π, then Uk satisfies
(4.3) and (PF ubini ) a.s. in Π and F0N . Assume that the initial condition uin
k , defined
1
k
k+1
as the distribution of (−T0 , A0 , . . . , A0 ) which is a random vector in R
, is such
k
)
=
0.
Then
U
admits
a
mean
measure
u
which
also
satisfies
({0}
×
R
that uin
k
k
+
k
,
(PF ubini ) and the following system in the weak sense: on R+ × Rk+1
+
∂
∂
+
uk (dt, ds, da1 , ..., dak )+f (s, a1 , ..., ak )uk (dt, ds, da1 , ..., dak ) = 0,
∂t ∂s
Z∞
uk (dt, 0, ds, da1 , ..., dak−1 ) =
f (s, a1 , ..., ak ) uk (t, ds, da1 , ..., dak ) dt,

(4.5)
(4.6)

ak =0

uk (0, ds, da1 , . . . , dak ) = uin
k (ds, da1 , . . . , dak ) .

(4.7)

k
We have assumed uin
k ({0}×R+ ) = 0 (i.e. T0 6= 0 a.s.) for the sake of simplicity,
but this assumption may of course be relaxed and Dirac masses at 0 should then
be added in a similar way as in Theorem 3.3.
If f ∈ L∞ (Rk+1
+ ), we may apply Proposition 4.1, so that the mean measure
k+1
1
uk satisfy System (4.5)–(4.7). Assuming an initial condition uin
k ∈ L (R+ ), we
can prove exactly as for the renewal equation (with a Banach fixed point argument for instance)
 34that there exists a unique solution uk such that (t 7→ uk (t, ·)) ∈
C R+ , L1 (Rk+1
)
to the generalized Wold case, the boundary assumption on the
+
kth penultimate point before time 0 being necessary to give sense to the successive
ages at time 0. By uniqueness, this proves
 that the mean measure uk is this solution,
so that it belongs to C R+ , L1 (Rk+1
: Proposition 4.1 leads to a regularity result
)
+
on the mean measure.

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Now that we have clarified the dynamics of the successive ages, one can look
at this system from the point of view of Theorem 3.3, that is when only two variables s and t are considered.
In this respect, let us note that U defined by (3.3) is
R
such that U (dt, ds) = a1 ,...,ak Uk (dt, ds, da1 , . . . , dak ). Since the integrals and the
expectations are exchangeable in the weak
R sense, the mean measure u defined in
Proposition 3.2 is such that u(dt, ds) = a1 ,...,ak uk (dt, ds, da1 , . . . , dak ). But (4.5)
∞
in the weak sense means, for all ϕ ∈ Cc,b
(Rk+2 ),

Z 
∂
∂
+
ϕ(t, s, a1 , ..., ak )uk (dt, ds, da1 , . . . , dak )
∂t ∂s
Z
+ [ϕ (t, 0, a1 , . . . , ak ) − ϕ(t, s, a1 , . . . , ak )] f (s, a1 , . . . , ak ) uk (dt, ds, da1 , . . . , dak )
Z
+ ϕ (0, s, a1 , . . . , ak ) uin
k (ds, da1 , . . . , dak ) = 0. (4.8)
∞
∞
Letting ψ ∈ Cc,b
(R2 ) and ϕ ∈ Cc,b
(Rk+2 ) being such that

∀ t, s, a1 , . . . , ak ,

ϕ(t, s, a1 , . . . , ak ) = ψ(t, s),

we end up proving that the function ρλ,P0 defined in Theorem 3.3 satisfies
Z
ρλ,P0 (t, s)u (dt, ds) =
f (s, a1 , . . . , ak ) uk (dt, ds, da1 , . . . , dak ) ,

(4.9)

a1 ,...,ak

u(dt, ds)−almost everywhere (a.e.). Equation (4.9) means exactly from a probabilistic point of view that


ρλ,P0 (t, s) = E f (St− , A1t , ..., Akt )|St− = s , u(dt, ds) − a.e.
Therefore, in the particular case of generalized Wold process, the quantity ρλ,P0
depends on the shape of the intensity (here the function f ) and also on uk . But,
by Proposition 4.1, uk depends on its initial condition given by the distribution of
(−T0 , A10 , . . . , Ak0 ), and not only −T0 as in the initial condition for u. That is, as
announced in the remarks following Theorem 3.3, ρλ,P0 depends in particular on the
whole distribution of the underlying process before time 0, namely P0 and not only
on the initial condition for u. Here, for generalized Wold processes, it only depends
on the last k points before time 0. For more general non Markovian settings, the
integration cannot be simply described by a measure uk in dimension (k + 2) being
integrated with respect to da1 ...dak . In general, the integration has to be done on
N
all the ”randomness” hidden behind the dependence of λ(t, Ft−
) with respect to
the past once St− is fixed and in this sense it depends on the whole distribution
P0 of N− . This is made even clearer on the following non Markovian example: the
Hawkes process.
4.3. Hawkes process
As we have seen in Section 2.2, there are many different
examples of Hawkes
proR


t−
N
cesses that can all be expressed as λ t, Ft− = φ −∞ h (t − x) N (dx) , where the
main case is φ(θ) = µ + θ, for µ some positive constant, which is the linear case.

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R


t−
N
When there is no point before 0, λ t, Ft−
= φ 0 h (t − x) N (dx) . In this
case, the interpretation is so close to (PPS) that the first guess, which is wrong,
would be that the analogous in (PPS) is
p(s, X(t)) = φ(X(t)),
(4.10)
hR
i
Rt
t−
where X(t) = E 0 h (t − x) N (dx) = 0 h (t − x) u(dx, 0). This is wrong, even

N
in the linear case since λ t, Ft−
depends on all the previous points. Therefore ρλ,P0
defined by (3.9) corresponds to a conditioning given only the last point.
By looking at this problem through the generalized Wold approach, one can
hope that for h decreasing fast enough:


N
λ t, Ft−
' φ h(St− ) + h(St− + A1t ) + ... + h(St− + A1t + ... + Akt ) .
In this sense and with respect to generalized Wold processes described in the
previous section, we are informally integrating on ”all the previous points” except
the last one and not integrating over all the previous points. This is informally
why (4.10) is wrong even in the linear case. Actually, ρλ,P0 is computableR for linear
t
Hawkes processes R: we show in the next section that ρλ,P0 (t, s) 6= φ( −∞ h(t −
∞
x)u(dx, 0)) = µ + 0 h(t − x)u(dx, 0) and that ρλ,P0 explicitly depends on P0 .
4.3.1. Linear Hawkes process
We are interested in Hawkes processes with a past before time 0 given by F0N , which
is not necessarily the past given by a stationary Hawkes process. To illustrate the
fact
 that
 the past is impacting the value of ρλ,P0 , we focus on two particular cases:

A1N
N− = {T0 } a.s. and T0 admits a bounded density f0 on R−
 −
A2N−
N− is an homogeneous Poisson process with intensity α on R−
Before stating the main result, we need some technical definitions. Indeed the
proof is based on the underlying branching structure of the linear Hawkes process
described in Section B.3.1 of the appendix and the following functions (Ls , Gs ) are
naturally linked to this branching decomposition (see Lemma B.7).
Lemma 4.2. Let h ∈ L1 (R+ ) such that khkL1 < 1. For all s ≥ 0, there exist a
unique solution (Ls , Gs ) ∈ L1 (R+ ) × L∞ (R+ ) of the following system
Z

(x−s)∨0

Z
Gs (x − w)h(w)dw −

log(Gs (x)) =
0

x

h(w)dw,

(4.11)

(h (w) + Ls (w)) Gs (w)h(x − w) dw,

(4.12)

0

Z

x

Ls (x) =
s∧x

where a ∨ b (resp. a ∧ b) denotes the maximum (resp. minimum) between a and b.
Moreover, Ls (x ≤ s) ≡ 0, Gs : R+ → [0, 1], and Ls is uniformly bounded in L1 .

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This result allows to define two other important quantities, Ks and q, by, for all
s, t ≥ 0, z ∈ R,
Z

(t−s)∨0

[h(t − x) + Ls (t − x)] Gs (t − x)h(x − z)dx,

Ks (t, z) :=
0

Z

t

log(q(t, s, z)) := −

Z

(t−s)∨0

h(x − z)dx −
(t−s)∨0

[1 − Gs (t − x)] h(x − z)dx. (4.13)
0

Finally, the following result is just an obvious remark that helps to understand the
resulting system.
Remark 4.1. For a non negative Φ ∈ L∞ (R+ ) and v in ∈ L∞ (R+ ), there exists a
unique solution v ∈ L∞ (R2+ ) in the weak sense to the following system,
∂
∂
v(t, s) +
v(t, s) + Φ(t, s)v(t, s) = 0,
∂t
∂s
v(t, 0) = 1
v(t = 0, s) = v in (s)

(4.14)
(4.15)

Moreover t 7→ v(t, .) is in C(R+ , L1loc (R+ )).
If v in is a survival function (i.e. non increasing from 0 to 1), then v(t, .) is a
survival function and −∂s v is a probability measure for all t > 0.
Proposition 4.3. Using the notations of Theorem
3.3,

 let
 N be
 a Hawkes process
with past before 0 given by N− satisfying either A1N− or A2N− and with intensity
on R+ given by
Z t−
N
λ(t, Ft−
)=µ+
h(t − x)N (dx),
−∞

where µ is a positive realRnumber and h ∈ L∞ (R+ ) is a non-negative function with
support in R+ such that h < 1.
Then, the mean measure u defined in Proposition 3.2 satisfies Theorem 3.3 and
R∞
moreover its integral v(t, s) := u(t, dσ) is the unique solution of the system (4.14)–
s
∞
(4.15) where v in is the survival function of −T0 , and where Φ = Φµ,h
P0 ∈ L (R+ ) is
defined by
µ,h
h
Φµ,h
P0 = Φ+ + Φ−,P0 ,

(4.16)

where for all non negative s, t

Z
Φµ,h
(t,
s)
=
µ
1
+
+

t


(h(x) + Ls (x))Gs (x)dx ,

(4.17)

s∧t



and where under Assumption A1N− ,
R 0∧(t−s)
Φh−,P0 (t, s)

=

−∞

(h(t − t0 ) + Ks (t, t0 )) q(t, s, t0 )f0 (t0 )dt0
,
R 0∧(t−s)
q(t, s, t0 )f0 (t0 )dt0
−∞

(4.18)

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or, under Assumption A2N− ,
Φh−,P0 (t, s) = α

Z

0∧(t−s)

(h(t − z) + Ks (t, z)) q(t, s, z)dz.

(4.19)

−∞

In these formulae, Ls , Gs , Ks and q are given by Lemma 4.2 and (4.13). Moreover
Z +∞
Z +∞
∀ s ≥ 0,
ρλ,P0 (t, x)u(t, dx) = Φµ,h
(t,
s)
u(t, dx).
(4.20)
P0
s

s

The proof is included in Appendix B.3. Proposition 4.3 givesa purely
 analytical


1
definition for v, and thus for u, in two specific cases, namely AN− or A2N− .
In the general case, treated in Appendix B (Proposition B.5), there remains a
dependence with respect to the initial condition P0 , via the function Φh−,P0 .
Remark 4.2. Contrarily to the general result in
Theorem 3.3, Proposition 4.3
R +∞
focuses on the equation satisfied by v(dt, s) = s u(dt, dx) because in Equation (4.14) the function parameter Φ = Φµ,h
P0 may be defined independently of the
definitions of v or u, which is not the case for the rate ρλ,P0 appearing in Equation (3.10). Thus, it is possible to depart from the system of equations defining v,
study it, prove existence, uniqueness and regularity for v under some assumptions
on the initial distribution uin as well as on the birth function h, and then deduce
regularity or asymptotic properties for u without any previous knowledge on the
underlying process.
In Sections 4.1 and 4.2, we were able to use the PDE formulation to prove that the
distribution of the ages u has a density. Here, since we only obtain a closed formula
for v and not for u, we would need to derive Equation (4.14) in s to obtain a similar
µ,h
result, so that we need to prove more regularity on Φµ,h
P0 . Such regularity for ΦP0
is not obvious since it depends strongly on the assumptions on N− . This paves the
way for future research, where the PDE formulation would provide regularity on
the distribution of the ages, as done above for renewal and Wold processes.




Remark 4.3. These two cases A1N− and A2N− highlight the dependence with
respect to all the past before time 0 (i.e. P0 ) and not only the initial condition (i.e.
in
the
 age at time 0). In fact, they can give the same initial condition u : for instance,
A1N−

with −T0 exponentially distributed with parameter α > 0 gives the same


law for −T0 as A2N− with parameter α. However, if we fix some non-negative real

number s, one can show that Φh−,P0 (0, s) is different in those two cases. It is clear
from the definitions that for every real number z, q(0, s, z) = 1 and Ks (0, z) = 0.
Thus, in the first case,
R −s
R∞
h(−t0 )αeαt0 dt0
h(z)αe−αz dz
−∞
h
Φ−,P0 (0, s) =
= s R ∞ −αz
,
R −s
αe
dz
αeαt0 dt0
s
−∞

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R −s
R∞
while in the second case, Φh−,P0 (0, s) = α −∞ h(−z)dz = α s h(w)dw. Therefore
Φh−,P0 clearly depends on P0 and not just on the distribution of the last point before
0, and so is ρλ,P0 .
Rt
Remark 4.4. If we follow our first guest, ρλ,P0 would be either µ + 0 h(t −
Rt
x)u(dx, 0) or µ+ −∞ h(t−x)u(dx, 0). In particular, it would not depend on the age
s. Therefore by (4.20), so would Φµ,h
P0 . But for instance at time t = 0, when N− is
R +∞
an homogeneous Poisson process of parameter α, Φµ,h
h(w)dw,
P0 (0, s) = µ + α s
which obviously depends on s. Therefore the intuition linking Hawkes processes and
(PPS) does not apply.
4.3.2. Linear Hawkes process with no past before time 0


A classical framework in point processes theory is the case in A1N− where T0 →
R t−
N
) = µ + 0 h(t − x)N (dx). The
−∞, or equivalently, when N has intensity λ(t, Ft−
problem in this case is that the age at time 0 is not finite. The age is only finite for
times greater than the first spiking time T1 .
Here again, the quantity v(t, s) reveals more informative and easier to use: having
the distribution of T0 going to −∞ means that Supp(uin ) goes to +∞, so that the
initial condition for v tends to value uniformly 1 for any 0 ≤ s < +∞. If we
can prove that the contribution of Φh−,P0 vanishes, the following system is a good
candidate to be the limit system:
∂ ∞
∂ ∞
∞
v (t, s) +
v (t, s) + Φµ,h
+ (t, s) v (t, s) = 0,
∂t
∂s
v ∞ (t, 0) = 1,
v ∞ (0, s) = 1,

(4.21)
(4.22)

where Φµ,h
+ is defined in Proposition 4.3. This leads us to the following proposition.
Proposition 4.4. Under the assumptions and notations of Proposition 4.3, consider for all M ≥ 0, vM theunique solution of system (4.14)-(4.15) with Φ given by
Proposition 4.3, case A1N− , with T0 uniformly distributed in [−M −1, −M ]. Then,

as M goes to infinity, vM converges uniformly on any set of the type (0, T ) × (0, S)
towards the unique solution v ∞ of System (4.21)-(4.22).
Conclusion
We present in this article a bridge between univariate point processes, that can
model the behavior of one neuron through its spike train, and a deterministic age
structured PDE introduced by Pakdaman, Perthame and Salort, named (PPS).
More precisely Theorem 3.3 present a PDE that is satisfied by the distribution u
of the age s at time t, where the age represents the delay between time t and the
last spike before t. This is done in a very weak sense and some technical structure,
namely (PF ubini ), is required.

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The main point is that the ”firing rate” which is a deterministic quantity written
as p(s, X(t)) in (PPS) becomes the conditional expectation of the intensity given
the age at time t in Theorem 3.3. This first makes clear that p(s, X(t)) should
be interpreted as a hazard rate, which gives the probability that a neuron fires
given that it has not fired yet. Next, it makes clearly rigorous several ”easy guess”
bridges between both set-ups when the intensity only depends on the age. But it
also explained why when the intensity has a more complex shape (Wold, Hawkes),
this term can keep in particular the memory of all that has happened before time 0.
One of the main point of the present study is the Hawkes process, for which
what was clearly expected was a legitimation of the term X(t) in the firing rate
p(s, X(t)) of (PPS), which models the synaptic integration. This is not the case,
and the interlinked equations that have been found for the cumulative distribution
function v(t, ·) do not have a simple nor direct deterministic interpretation. However
one should keep in mind that the present bridge, in particular in the population wide
approach, has been done for independent neurons. This has been done to keep the
complexity of the present work reasonable as a first step. But it is also quite obvious
that interacting neurons cannot be independent. So one of the main question is: can
we recover (PPS) as a limit with precisely a term of the form X(t) if we consider
multivariate Hawkes processes that really model interacting neurons ?

Acknowledgment
This research was partly supported by the European Research Council (ERC Starting Grant SKIPPERAD number 306321), by the french Agence Nationale de la
Recherche (ANR 2011 BS01 010 01 projet Calibration) and by the interdisciplanary axis MTC-NSC of the University of Nice Sophia-Antipolis. MJC acknowledges
support from the projects MTM2014-52056-P (Spain) and P08-FQM-04267 from
Junta de Andalucı́a (Spain). We warmly thank François Delarue for very fruitful
discussions and ideas.

A. Proofs linked with the PDE
A.1. Proof of Proposition 3.1
First, let us verify that U satisfies Equation (3.2). For any t > 0,

U (t, ds) =

X

ηTi (t, ds)10≤t≤Ti+1 ,

i≥0

by definition of U . Yet, ηTi (t, ds) = δt−Ti (ds)1t>Ti , and the only i ∈ N such that
Ti < t ≤ Ti+1 is i = Nt− . So, for all t > 0, U (t, ds) = δt−TNt− (ds) = δSt− (ds).
Secondly, let us verify that U satisfies (PF ubini ). Let ϕ ∈ Mc,b (R2+ ), and let T be

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(1)

P+∞
= 0. Then since U (t, ds) = i=0 ηTi (t, ds)10≤t≤Ti+1 ,


!
Z
Z
X

ϕ(t, s)U (t, ds) dt ≤
|ϕ(t, s)|
ηTi (t, ds)10≤t≤Ti+1  dt

such that for all t > T , ϕt
Z

Z

R+

=

R+

R+

XZ
i≥0

R+

|ϕ(t, t − Ti )|1t>Ti 10≤t≤Ti+1 dt =

R+

i≥0

XZ
i≥0

Z

T1

0

|ϕ(t, t − Ti )|dt

max(0,Ti )

Z

X

|ϕ(t, t − T0 )| +

=

Ti+1

Ti+1

|ϕ(t, t − Ti )|dt.

i/0<Ti <T

Ti

Since there is a finite number of points of NR between
0 and T , on Ω, this quantity
P
+∞ R +∞
is finite and one can exchange i≥0 and t=0 s=0 . Therefore, since all the ηTi
satisfy (PF ubini ) and ϕ(t, s)10≤t≤Ti+1 is in Mc,b (R2+ ), so does U .
∞
For the dynamics of U , similar computations lead for every ϕ ∈ Cc,b
(R+ 2 ) to
Z
X Z Ti+1 −Ti
ϕ (t, s) U (dt, ds) =
ϕ (s + Ti , s) ds.
i≥0

max(0,−Ti )

We also have


Z 
X Z Ti+1 −Ti  ∂
∂
∂
∂
+
ϕ (t, s) U (dt, ds) =
+
ϕ (s + Ti , s) ds
∂t ∂s
∂t ∂s
i≥0 max(0,−Ti )
X
=
[ϕ (Ti+1 , Ti+1 − Ti ) − ϕ (Ti , 0)] + ϕ(T1 , T1 − T0 ) − ϕ(0, −T0 ).
(A.1)
i≥1

R λ(t,F N )
P
It remains to express the term with x=0 t− Π (dt, dx) = i≥0 δTi+1 (dt), that is
X
Z
Z Z
X
ϕ (t, s) U (t, ds)
δTi+1 (dt) =
ϕ (t, s) U (t, ds)
δTi+1 (dt)
i≥0

i≥0

Z
=

ϕ (t, St− )

X
i≥0

and, since

R

δTi+1 (dt) =

X

ϕ (Ti+1 , Ti+1 − Ti ) , (A.2)

i≥0

U (t, ds) = 1 for all t > 0,
Z Z
X
X
ϕ (t, 0) U (t, ds)
δTi+1 (dt) =
ϕ (Ti+1 .0) ,
i≥0

(A.3)

i≥0

Identifying all the terms in the right-hand side of Equation (A.1), this lead to
Equation (3.7), which is the weak formulation of System (3.4)–(3.6).
A.2. Proof of Proposition 3.2
(1)

Let ϕ ∈ Mc,b (R2+ ), and let T be such that for all t > T , ϕt = 0. Then,
Z
|ϕ(t, s)|U (t, ds) ≤ ||ϕ||L∞ 10≤t≤T ,

(A.4)

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R
since
R at any fixed time t > 0, U (t, ds) = 1. Therefore, the expectation
E ϕ(t, s)U (t, ds) is well-defined and finite and so u(t, .) is well-defined.
On the other hand, at any fixed age s,
Z
∞
X
|ϕ(t, s)|U (dt, s) =
|ϕ(s + Ti , s)|10≤s≤Ti+1 −Ti
i=0

=

X

|ϕ(s + Ti , s)|10≤s+Ti ≤T 10≤s≤Ti+1 −Ti ,

i≥0
(1)

because for all t > T , ϕt
Z
|ϕ(t, s)|U (dt, s)

= 0. Then, one can deduce the following bound

≤ |ϕ(s + T0 , s)|1−T0 ≤s≤T −T0 10≤s≤T1 −T0 +

X

|ϕ(s + Ti , s)|10≤s≤T 1Ti ≤T

i≥1

Since the intensity is L1loc
Z
E

≤ ||ϕ||L∞ (1−T0 ≤s≤T −T0 + NT 10≤s≤T ) .
hR
i
T
N
in expectation, E [NT ] = E 0 λ(t, Ft−
)dt <∞ and


|ϕ(t, s)|U (dt, s) ≤ ||ϕ||L∞ (E [1−T0 ≤s≤T −T0 ] + E [NT ] 10≤s≤T ) ,

(A.5)

so the expectation is well-defined and finite and so u(·, s) is well-defined.
Now, let us show (PF ubini ). First Equation (A.4) implies
Z

Z
E
|ϕ(t, s)|U (t, ds) dt ≤ T ||ϕ||L∞ ,
and Fubini’s theorem implies that the following integrals are well-defined and that
the following equality holds,
Z

Z Z

Z
E
ϕ(t, s)U (t, ds) dt = E
ϕ(t, s)U (t, ds)dt .
(A.6)
Secondly, Equation (A.5) implies
Z

Z
E
|ϕ(t, s)|U (dt, s) ds ≤ ||ϕ||L∞ (T + T E [NT ]) ,
by exchanging the integral with the expectation and Fubini’s theorem implies that
the following integrals are well-defined and that the following equality holds,
Z

Z Z

Z
E
ϕ(t, s)U (dt, s) ds = E
ϕ(t, s)U (dt, s)ds .
(A.7)
Now, it only remains to use (PF ubini ) for U to deduce that the right members of
Equations (A.6) and (A.7)
Rare
 (PF ubini ) for U tells that these two
R equal. Moreover,
quantities are equal to E
ϕ(t, s)U (dt, ds) . This concludes the proof.

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A.3. Proof of Theorem 3.3


N
E λ(t,Ft−
)1|S

t− −s|≤ε

29



, for every t > 0 and s ≥ 0. Since
Let ρλ,P0 (t, s) := lim inf ε↓0
P(|St− −s|≤ε)
N
(λ(t, Ft− ))t>0 and (St− )t>0 are predictable processes, and a fortiori progressive
processes (see page 9 in 3 ), ρλ,P0 is a measurable function of (t, s).

N
For every t > 0, let µt be the measure defined by
µ
(A)
=
E
λ(t,
F
)1
(S
)
t
A
t−
t−
 1

,exp
for all measurable set A. Since Assumption ALλ,loc
implies that dt-a.e.


N
E λ(t, Ft− ) < ∞ and since u(t, ds) is the distribution of St− , µt is absolutely
continuous with respect to u(t, ds) for dt-almost every t.
Let ft denote the Radon Nikodym derivative
 of µt with respect to u(t, ds).
N
) St− = s by definition of the conditional
For u(t, ds)-a.e. s, ft (s) = E λ(t, Ft−
expectation. Moreover, a Theorem of Besicovitch 27 claims
that for u(t, ds)-a.e.


N
) St− = s holds
s, ft (s) = ρλ,P0 (t, s). Hence, the equalityρλ,P0 (t, s) = E λ(t, Ft−
u(t, ds)dt = u(dt, ds)-almost everywhere.
Next, in order to use (PF ubini ), let us note that for any T, K > 0,
2
ρK,T
(A.8)
λ,P0 : (t, s) 7→ (ρλ,P0 (t, s) ∧ K) 10≤t≤T ∈ Mc,b (R+ )

R R K,T
R R K,T
Hence,
ρλ,P0 (t, s)u(dt, ds) =
ρλ,P0 (t, s)u(t, ds) dt which is always upper


RT R
RT
RT 
N
bounded by 0
ρλ,P0 (t, s)u(t, ds) dt = 0 µt (R+ )dt = 0 E λ(t, Ft−
) dt < ∞.
RT R
Letting K → ∞, one has that 0 ρλ,P0 (t, s)u(dt, ds) is finite for all T > 0.
Once ρλ,P0 correctly defined, the proof of Theorem 3.3 is a direct consequence
of Proposition 3.1.
More precisely, let us show that (3.7) implies (3.13). Taking the expectation
∞
of (3.7) gives that for all ϕ ∈ Cc,b
(R2+ ),
N
Z λ(t,Ft−
)

"Z
E

[ϕ (t, s) − ϕ(t, 0)]

!

#

Z

Π (dt, dx) U (t, ds) −

ϕ (0, s) uin (ds)

x=0

Z
−

(∂t + ∂s ) ϕ (t, s) u (dt, ds) = 0. (A.9)

denote ψ(t,s) := ϕ(t, s) − ϕ(t, 0). Due to Ogata’s thinning construction,
 Let us
N
R λ(t,Ft−
)
Π (dt, dx) = N (dt)1t>0 where N is the point process constructed by
x=0
thinning, and so,
!
#
"Z
N
Z
Z λ(t,Ft−
)
E
ψ (t, s)
Π (dt, dx) U (t, ds) = E
x=0


ψ (t, St− ) N (dt) .

t>0

(A.10)
But ψ(t, St− ) is a (FtN )-predictable process and
Z
E
t>0

N
|ψ(t, St− )|λ(t, Ft−
)dt

"Z


≤ kψkL∞ E

0

#

T
N
λ(t, Ft−
)dt

< ∞,

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hence, using the martingale property of the predictable intensity,
Z

Z


N
E
ψ (t, St− ) N (dt) = E
ψ (t, St− ) λ t, Ft−
dt .
t>0

(A.11)

t>0

Moreover,
thanks to Fubini’s Theorem, the right-hand term is finite and equal to
R
N
E[ψ (t, St− ) λ(t, Ft−
)]dt, which can also be seen as
Z
Z
E [ψ (t, St− ) ρλ,P0 (t, St− )] dt = ψ(t, s)ρλ,P0 (t, s)u(t, ds)dt.
(A.12)
For all K > R0, ((t, s) 7→ ψ(t, s) (ρλ,P0 (t, s) ∧ K)) ∈RMc,b (R2+ ) and, from (PF ubini ), it
is clear that ψ(t, s) (ρλ,P0 (t, s) ∧ K) u(t, ds)dt = ψ(t, s) (ρλ,P0 (t, s) ∧ K) u(dt, ds).
Since RoneR can always upper-bound this quantity in absolute value by
T
kψkL∞ 0 s ρλ,P0 (t, s)u(dt, ds), this is finite. Letting K → ∞ one can show that
Z
Z
ψ(t, s)ρλ,P0 (t, s)u(t, ds)dt = ψ(t, s)ρλ,P0 (t, s)u(dt, ds).
(A.13)
Gathering (A.10)-(A.13) with (A.9) gives (3.13).
A.4. Proof of Corollary 3.4
i
i
For all i ∈ N∗ , let us denote N+
= N i ∩ (0, +∞) and N−
= N i ∩ R− . Thanks
i
can be seen as constructed via thinning
to Proposition B.12, the processes N+
2
of independent Poisson processes on R+ . Let (Πi )i∈N be the sequence of point
measures associated to independent Poisson processes of intensity 1 on R2+ given
i
. In particular,
by Proposition B.12. Let T0i denote the closest point to 0 in N−
i
(T0 )i∈N∗ is a sequence of i.i.d. random variables.
For each i, let U i denote the solution of the microscopic equation corresponding
to Πi and T0i as defined in Proposition 3.1 by (3.3). Using (3.2), it is clear that
Pn
Pn
i
∞
2
i (ds) =
i=1 δSt−
i=1 U (t, ds) for all t > 0. Then, for every ϕ ∈ Cc,b (R+ ),
!
Z
n
n Z
1X
1X
δ i (ds) =
ϕ(t, s)U i (t, ds).
ϕ(t, s)
n i=1 St
n i=1
R
The right-hand side is a sum n i.i.d. random variables with mean ϕ(t, s)u(t, ds),
so (3.14) clearly follows from the law of large numbers.

B. Proofs linked with the various examples
B.1. Renewal process
Proposition B.1. With the notations of Section 2, let N be a point process on R,
with predictable age process (St− )t>0 , such that T0 = 0 a.s. The following statements
are equivalent:
(i) N+ = N ∩ (0, +∞) is a renewal process with ISI’s distribution given by some
density ν : R+ → R+ .

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31


N
N
(ii) N admits λ(t, Ft−
) = f (St− ) as an intensity on (0, +∞) and λ(t, Ft−
) t>0
 1

,a.s.
satisfies ALλ,loc
, for some f : R+ → R+ .
In such a case, for all x ≥ 0, f and ν satisfy
Z x
• ν(x) = f (x) exp(−
f (y)dy) with the convention exp(−∞) = 0,
0
Z ∞
ν(x)
R
• f (x) = ∞
if
ν(y)dy 6= 0, else f (x) = 0.
ν(y)dy
x
x

(B.1)
(B.2)

Proof. For (ii) ⇒ (i). Since T0 = 0 a.s., Point (2) of Proposition B.2 given later
on for the general Wold case implies that the ISI’s of N forms a Markov chain of
order 0 i.e. they are i.i.d. with density given
R ∞ by (B.1).
For (i) ⇒ (ii). Let x0 = inf{x ≥ 0, x ν(y)dy = 0}. It may be infinite. Let us
define f by (B.2) for every 0 ≤ x < x0 and let Ñ be a point process on R such
Ñ
Ñ
that Ñ− = N− and Ñ admits λ(t, Ft−
) = f (St−
) as an intensity on (0, +∞) where
Ñ
(St− )t>0 is the predictable age process associated to Ñ . Applying (ii) ⇒ (i) to Ñ
gives that the ISI’s of Ñ are i.i.d. with density given by
!
Z x
ν(x)
ν(y)
R∞
ν̃(x) = R ∞
exp −
dy ,
ν(y)dy
ν(z)dz
0
x
y
for every 0 ≤ x < x
0 and ν̃(x) = 0 for x
≥ x0 . It is clear that ν = ν̃ since the function
Rx
ν(y)
1
exp − 0 R ∞
dy is differentiable with derivative equal to 0.
x 7→ R ∞
x

ν(y)dy

y

ν(z)dz

Since N and Ñ are renewal processes with same density ν and same first point
T0 = 0, they have the same distribution. Since the intensity characterizes a point
N
N
process, N also admits λ(t, Ft−
) = f (St−
) as an intensity on (0, +∞). Moreover,

N
since N is a renewal process, it is non-explosive in finite time and so λ(t, Ft−
) t>0
 1

,a.s.
satisfies ALλ,loc
.

B.2. Generalized Wold processes
In this Section, we suppose that there exists k ≥ 0 such that the underlying point
process N has intensity

N
λ t, Ft−
= f (St− , A1t , ..., Akt ),
(B.3)
where f is a function and the Ai ’s are defined by Equation (2.2).
B.2.1. Markovian property and the resulting PDE
Let N be a point process of intensity given by (B.3). If T−k > −∞, its associated
age process (St )t can be defined up to t > T−k . Then let, for any integer i ≥ −k,
Ai = Ti+1 − Ti = STi+1 −

(B.4)

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and denote (FiA )i≥−k the natural filtration associated to (Ai )i≥−k .
For any t ≥ 0, and point process Π on R2+ , let us denote Π≥t (resp. Π>t ) the
restriction to R2+ (resp. (0, +∞) × R+ ) of the point process Π shifted t time units
to the left on the first coordinate. That is, Π≥t (C × D) = Π((t + C) × D) for all
C ∈ B(R+ ), D ∈ B(R+ ) (resp. C ∈ B((0, +∞))).
Proposition B.2.
Let consider k a non-negative integer, f some non negative function on Rk+1
+
and N a generalized Wold process of intensity given by (B.3).Supposethat P0 is

1
,a.s.
N
) t>0 satisfies ALλ,loc
. Then,
such that P0 (T−k > −∞) = 1 and that λ(t, Ft−

(1) If (Xt )t≥0 = (St− , A1t , ..., Akt ) t≥0 , then for any finite non-negative stopping time τ , (Xtτ )t≥0 = (Xt+τ )t≥0 is independent of FτN− given Xτ .
(2) the process (Ai )i≥1 given by (B.4) forms a Markov chain of order k with
transition measure given by
 Z x

ν(dx, y1 , ..., yk ) = f (x, y1 , ..., yk ) exp −
f (z, y1 , ..., yk )dz dx. (B.5)
0

If T0 = 0 a.s., this holds for (Ai )i≥0 .
If f is continuous then G, the infinitesimal generator of (Xt )t≥0 , is given by
∀φ ∈ C 1 (Rk+1
(Gφ)(s, a1 , ..., ak ) =
+ ),
∂
φ(s, a1 , ..., ak ) + f (s, a1 , ..., ak ) (φ(0, s, a1 , ..., ak−1 ) − φ(s, a1 , ..., ak )) . (B.6)
∂s
Proof. First, let us show the first point of the Proposition. Let Π be such that N
is the process resulting of Ogata’s thinning with Poisson measure Π. The existence
of such a measure is assured by Proposition B.12. We show that for any finite
stopping time τ , the process (Xtτ )t≥0 can be expressed as a function of Xτ and
Π≥τ which is the restriction to R2+ of the Poisson process Π shifted τ time units
to the left on the first coordinate. Let e1 = (1, 0, . . . , 0) ∈ Rk+1 . For all t ≥ 0, let
Yt = Xτ + te1 and define
(
)
Z
Z
f (Yw )

R0 = inf

t ≥ 0,

Π≥τ (dw, dx) = 1 .
[0,t]

x=0

Note that R0 may be null, in particular when τ is a jumping time of the underlying
point process N . It is easy to check that R0 can be expressed as a measurable
τ
function of Xτ and Π≥τ . Moreover, it is clear that Xt∧R
= Yt∧R0 for all t ≥ 0.
0
So, R0 can be seen as the delay until the first point of the underlying process N
after time τ . Suppose that Rp , the delay until the (p + 1)th point, is constructed
for some p ≥ 0 and let us show how Rp+1 can be constructed. For t ≥ Rp , let
τ
Zt = θ(XR
)+te1 , where θ : (x1 , . . . , xk+1 ) 7→ (0, x1 , . . . , xk ) is a right shift operator
p

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33

modelling the dynamics described by (2.3). Let us define
(
Rp+1 = inf

Z

Z

f (Zw )

t > Rp ,

)
Π≥τ (dw, dx) = 1 .

(Rp ,Rp +t]

(B.7)

x=0

Note that for any p ≥ 0, Rp+1 cannot be null. It is coherent with the fact that the
counting process (Nt )t>0 only admits jumps with height 1. It is easy to check that
τ
Rp+1 can be expressed as a measurable function of θ(XR
) and Π>τ +Rp . It is also
p
τ
clear that Xt∧Rp+1 = Zt∧Rp+1 for all t ≥ Rp . So, Rp+1 can be seen as the delay
τ
can be
until the (p + 2)th point of the process N after time τ . By induction, XR
p
τ
expressed as a function of Xτ and Π≥τ , and this holds for Rp+1 and XRp+1 too.
To conclude, remark that the process (Xtτ )t≥0 is a measurable function of Xτ
and all the Rp ’s for p ≥ 0. Thanks to the independence of the Poisson measure
Π, FτN− is independent of Π≥τ . Then, since (Xtτ )t≥0 is a function of Xτ and Π≥τ ,
(Xtτ )t≥0 is independent of FτN− given Xτ which concludes the first point.
For Point (2), fix i ≥ 1 and apply Point (1) with τ = Ti . It appears that
in this case, R0 = 0 and R1 = Ai . Moreover, R1 = Ai can be expressed as a
A
function of θ(Xτ ) and Π>τ . However, θ(Xτ ) = (0, Ai−1 , . . . , Ai−k ) and Fi−1
⊂ FTNi .
A
given
Since τ = Ti , Π>τ is independent of FTNi and so Ai is independent of Fi−1
(Ai−1 , . . . , Ai−k ). That is, (Ai )i≥1 forms a Markov chain of order k.
Note that if T0 = 0 a.s. (in particular it is non-negative), then one can use the
previous argumentation with τ = 0 and conclude that the Markov chain starts one
time step earlier, i.e. (Ai )i≥0 forms a Markov chain of order k.
For (B.5),R1 = Ai , defined by (B.7), has the same distribution as the first
point of a Poisson process with intensity λ(t) = f (t, Ai−1 , . . . , Ai−k ) thanks to the
thinning Theorem. Hence, the transition measure of (Ai )i≥1 is given by (B.5).
Now that (Xt )t≥0 is Markovian, one can compute its infinitesimal generator.
Suppose that f is continuous and let φ ∈ Cb1 (Rk+1
+ ), The generator of (Xt )t≥0 is
defined by Gφ(s, a1 , . . . , ak ) = limh→0+ Phh−Id φ, where
Ph φ (s, a1 , . . . , ak ) = E [φ (Xh )|X0 = (s, a1 , . . . , ak )]


= E φ (Xh ) 1{N ([0,h])=0} X0 = (s, a1 , . . . , ak )


+E φ (Xh ) 1{N ([0,h])>0} X0 = (s, a1 , . . . , ak )
= E0 + E>0 .
The case with no jump is easy to compute,
E0 = φ (s + h, a1 , . . . , ak ) (1 − f (s, a1 , . . . , ak ) h) + o(h),

(B.8)

thanks to the continuity of f . When h is small, the probability to have more than
two jumps in [0, h] is a o(h), so the second case can be reduced to the case with

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exactly one random jump (namely T ),


E>0 = E φ (Xh ) 1{N ([0,h])=1} X0 = (s, a1 , . . . , ak ) + o(h)


= E φ (θ(X0 + T ) + (h − T )e1 ) 1{N ∩[0,h]={T }} X0 = (s, a1 , . . . , ak ) + o(h)


= E (φ (0, s, a1 , . . . , ak−1 ) + o(1)) 1{N ∩[0,h]={T }} X0 = (s, a1 , . . . , ak ) + o(h)
= φ (0, s, a1 . . . , ak−1 ) (f (s, a1 , . . . , ak ) h) + o (h) ,

(B.9)

thanks to the continuity of φ and f . Gathering (B.8) and (B.9) with the definition
of the generator gives (B.6).

B.2.2. Sketch of proof of Proposition 4.1
Let N be the point process construct by Ogata’s thinning of the Poisson process Π
and Uk be as defined in Proposition 4.1. By an easy generalisation of Proposition
3.1, one can prove that on the event Ω of probability 1, where Ogata’s thinning is
well defined, and where T0 < 0, Uk satisfies (PF ubini ), (4.3) and on R+ × Rk+1
+ , the
following system in the weak sense
!


Z f (s,a1 ,...,ak )
∂
∂
+
Uk (dt, ds, da) +
Π (dt, dx) Uk (t, ds, da) = 0,
∂t ∂s
x=0
!
Z
Z
f (s,a1 ,...,ak )

Uk (dt, 0, ds, da1 , ..., dak−1 ) =

Π (dt, dx) Uk (t, ds, da) ,
ak ∈R

x=0

with da = da1 × ... × dak and initial condition U in = δ(−T0 ,A10 ,...,Ak0 ) .
Similarly to R
Proposition 3.2, one can
ϕ in
 also prove
R that for any test function

Mc,b (Rk+2
),
E
ϕ(t,
s,
a)U
(t,
ds,
da)
and
E
ϕ(t,
s,
a)U
(dt,
s,
da)
are
finite
k
k
+
and one can define uk (t, ds, da) and uk (dt, s, da) by, for all ϕ in Mc,b (Rk+2
+ ),
Z

Z
ϕ(t, s, a)uk (t, ds, da) = E
ϕ(t, s, a)Uk (t, ds, da) ,
for all t ≥ 0, and
Z

Z
ϕ(t, s, a)uk (dt, s, da) = E
ϕ(t, s, a)Uk (dt, s, da) ,
for all s ≥ 0. Moreover, uk (t, ds, da) and uk (dt, s, da) satisfy (PF ubini ) and one can
define uk (dt, ds, da) = uk (t, ds, da)dt = uk (dt, s, da)ds on R2+ , such that for any
test function ϕ in Mc,b (Rk+2
+ ),
Z

Z
ϕ(t, s, a)uk (dt, ds, da) = E
ϕ(t, s, a)Uk (dt, ds, da) ,
quantity which is finite. The end of the proof is completely analogous to the one of
Theorem 3.3.

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B.3. Linear Hawkes processes
B.3.1. Cluster decomposition
Proposition B.3. Let g be a non negative L1loc (R+ ) function and h a non negative
L1 (R+ ) function such that khk1 < 1. Then the branching point process N is defined
as ∪∞
k=0 Nk the set of all the points in all generations constructed as follows:
• Ancestral points are Nanc distributed as a Poisson process of intensity g;
N0 := Nanc can be seen as the points of generation 0.
• Conditionally to Nanc , each ancestor a ∈ Nanc gives birth, independently
of anything else, to children points N1,a according to a Poisson process of
intensity h(. − a); N1 = ∪a∈Nanc N1,a forms the first generation points.
Then the construction is recursive in k, the number of generations:
• Denoting Nk the set of points in generation k, then conditionally to Nk ,
each point x ∈ Nk gives birth, independently of anything else, to children
points Nk+1,x according to a Poisson process of intensity h(. − x); Nk+1 =
∪x∈Nk Nk+1,x forms the points of the (k + 1)th generation.
This construction ends almost surely in every finite interval. Moreover the intensity
of N exists and is given by
Z t−
N
λ(t, Ft−
) = g(t) +
h(t − x)N (dx).
0

This is the cluster representation of the Hawkes process. When g ≡ ν, this has
been proved in 20 . However up to our knowledge this has not been written for a
general function g.
Proof. First, let us fix some A > 0. The process ends up almost surely in [0, A]
because there is a.s. a finite number of ancestors in [0, A]: if we consider the family of
points attached to one particular ancestor, the number of points in each generation
form a sub-critical Galton
Watson process with reproduction distribution, a Poisson
R
variable with mean h < 1 and whose extinction is consequently almost sure.
Next, to prove that N has intensity
Z t−
H(t) = g(t) +
h(t − x)N (dx),
0

we exhibit a particular thinning construction, where on one hand, N is indeed a
branching process by construction as defined by the proposition and, which, on the
other hand, guarantees that Ogata’s thinning project the points below H(t). We
can always assume that h(0) = 0, since changing the intensity of Poisson process
in the Rbranching structure at one particular point has no impact. Hence H(t) =
t
g(t) + 0 h(t − x)N (dx).
The construction is recursive in the same way. Fix some realisation Π of a Poisson
process on R2+ .

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For Nanc , project the points below the curve t → g(t) on [0, A]. By construction,
Nanc is a Poisson process of intensity g(t) on [0, A]. Note that for the identification
(see Theorem B.11) we just need to do it on finite intervals and that the ancestors
that may be born after time A do not have any descendants in [0, A], so we can
discard them, since they do not appear in H(t), for t ≤ A.
Enumerate the points in Nanc ∩ [0, A] from T1 to TN0,∞ .
• The children of T1 , N1,T1 , are given by the projection of the points of Π
whose ordinates are in the strip t 7→ (g(t), g(t)+h(t−T1 )]. As before, by the
property of spatial independence of Π, this is a Poisson process of intensity
h(. − T1 ) conditionally to Nanc .
• Repeat until TN0,∞ , where N1,TN0,∞ are given by the projection of the points
PN0,∞ −1
of Π whose ordinates are in the strip t 7→ (g(t) + i=1
h(t − Ti ), g(t) +
PN0,∞
h(t
−
T
)].
As
before,
by
the
property
of
independence
of Π, this is a
i
i=1
Poisson process of intensity h(. − TN0,∞ ) conditionally to Nanc and because
the consecutive strips do not overlap, this process is completely independent
of the previous processes (N1,Ti )’s that have been constructed.
Note that at the end of this first generation, N1 = ∪T ∈Nanc N1,T consists of the
PN0,∞
projection of points of Π in the strip t 7→ (g(t), g(t) + i=1
R h(t − Ti )]. They
PN0,∞
therefore form a Poisson process of intensity i=1 h(t − Ti ) = h(t − u)Nanc (du),
conditionally to Nanc .
For generation
k + 1 replace in the previous construction Nanc by Nk and g(t)
Pk−1 R
by g(t) + j=0 h(t − u)dNj (u). Once again we end up for each point x in Nk
with a process of children Nk+1,x which is a Poisson process of intensity h(t − x)
conditionally to Nk and which is totally independent of the other Nk+1,y ’s.
R Note
also that as before, Nk+1 = ∪x∈Nk Nk+1,x is a Poisson process of intensity h(t −
u)Nk (du), conditionally to N0 , ..., Nk .
Hence we are indeed constructing a branching process as defined by the proposition. Because the underlying Galton Watson process ends almost surely, as shown
before, it means that there exists a.s. one generation Nk∗ which will be completely
empty and our recursive contruction ends up too.
The main point is to realize that at the end the points in N = ∪∞
k=0 Nk are
exactly the projection of the points in Π that are below
∞ Z
∞ Z t
X
X
t 7→ g(t) +
h(t − u)Nk (du) = g(t) +
h(t − u)Nk (du)
k=0

k=0

0

hence below
Z
t 7→ g(t) +

t

h(t − u)N (du) = H(t).
0

Moreover H(t) is FtN predictable. Therefore by Theorem B.11, N has intensity
H(t), which concludes the proof.

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A cluster process Nc , is aR branching process, as defined before, which admits
t−
Nc
intensity λ(t, Ft−
) = h(t) + 0 h(t − z)Nc (dz). Its distribution only depends on
the function h. It corresponds to the family generated by one ancestor at time 0 in
Proposition B.3. Therefore, by Proposition
B.3, a Hawkes process with empty past
R t−
N
(N− = ∅) of intensity λ(t, Ft−
) = g(t) + 0 h(t − z)N (dz) can always be seen as
the union of Nanc and of all the a + Nca for a ∈ Nanc where the Nca are i.i.d. cluster
processes.
For a Hawkes process N with non empty past, N− , this
 is more technical. Let
Nanc be a Poisson process of intensity g on R+ and NcV V ∈Nanc be a sequence of
i.i.d. cluster processes associated to h. Let also
!
[
V
N>0 = Nanc ∪
V + Nc .
(B.10)
V ∈Nanc

As we prove below, this represents the points in N that do not depend on N− . The
points that are depending
on N− are constructed as follows independently of N>0 .

T
Given N− , let N1 T ∈N− denote a sequence of independent Poisson processes with

respective intensities λT (v) = h(v − T )1(0,∞) (v). Then, given N− and N1T T ∈N− ,

let NcT,V V ∈N T ,T ∈N be a sequence of i.i.d. cluster processes associated to h. The
−
1
points depending on the past N− are given by the following formula as proved in
the next Proposition:



[
[
N≤0 = N− ∪ 
N1T ∪ 
V + NcT,V  .
(B.11)
T ∈N−

V ∈N1T

Proposition B.4. Let N = N≤0 ∪ N>0 , where N>0 and N≤0 are given by (B.10)
and (B.11). Then N is a linear HawkesR process with past given by N− and intensity
t−
N
) = g(t) + −∞ h(t − x)N (dx), where g and h are as in
on (0, ∞) given by λ(t, Ft−
Proposition B.3.
Proof. Proposition B.3 yields that N>0 has intensity
Z t−
N>0
λN>0 (t, Ft−
) = g(t) +
h(t − x)N>0 (dx),

(B.12)

0
T
and that, given N− , for any T ∈ N− , NH
= N1T ∪
NT
λNHT (t, Ft−H )

S

V

∈N1T

V + NcT,V



has intensity

t−

Z

T
h(t − x)NH
(dx),

= h(t − T ) +

(B.13)

0

Moreover, all these processes are
Windependent
 given N− . For any t ≥ 0, one can
T
N≤0
NH
N−
note that Ft
⊂ Gt := F0 ∨
, and so N≤0 has intensity
T ∈N− Ft
λN≤0 (t, Gt− ) =

X
T ∈N−

NT
λNHT (t, Ft−H )

Z

t−

h(t − x)N≤0 (dx)

=
−∞

(B.14)

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N

on (0, +∞). Since this last expression is Ft ≤0 -predictable, by page 27 in 3 , this is
N
also λN≤0 (t, Ft−≤0 ). Moreover, N≤0 and N>0 are independent by construction and,
N≤0

for any t ≥ 0, FtN ⊂ Ft
given by

∨ FtN>0 . Hence, as before, N has intensity on (0, +∞)

N

N>0
N
) = g(t) +
λ(t, Ft−
) = λ(t, Ft−≤0 ) + λ(t, Ft−

Z

t−

h(t − x)N (dx).
−∞

B.3.2. A general result for linear Hawkes processes
The following proposition is a consequence of Theorem 3.3 applied to Hawkes processes with general past N− .
Proposition B.5. Using the notations of Theorem 3.3, let N be a Hawkes process
with past before 0 given by N− of distribution P0 and with intensity on R+ given by
Z t−
N
λ(t, Ft− ) = µ +
h(t − x)N (dx),
−∞

where µ is a positive
real number and h is a non-negative function with support in
R
R+ such that h < 1. Suppose that P0 is such that
Z 0

sup E
h(t − x)N− (dx) < ∞.
(B.15)
t≥0

−∞

Then, the mean measure u defined in Proposition 3.2 satisfies Theorem 3.3 and
R∞
moreover its integral v(t, s) := u(t, dσ) is a solution of the system (4.14)–(4.15)
s
µ,h
where v in is the survival function of −T0 , and where Φ = Φµ,h
P0 is given by ΦP0 =
µ,h
µ,h
µ,h
Φµ,h
+ + Φ−,P0 , with Φ+ given by (4.17) and Φ−,P0 given by,
Z t−

∀ s, t ≥ 0, Φµ,h
(t,
s)
=
E
h(t
−
z)N
(dz)
N
([t
−
s,
t))
=
0
. (B.16)
≤0
≤0
−,P0
−∞

Moreover, (4.20) holds.
B.3.3. Proof of the general result of Proposition B.5
Before proving Proposition B.5, we need some technical preliminaries.
Events of the type {St− ≥ s} are equivalent to the fact that the underlying
process has no point between t − s and t. Therefore, for any point process N and
any real numbers t, s ≥ 0, let
Et,s (N ) = {N ∩ [t − s, t) = ∅}.

(B.17)

Various sets Et,s (N ) are used in the sequel and the following lemma, whose proof is
obvious and therefore omitted, is applied several times to those sets.

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Lemma B.6. Let Y be some random variable and I(Y ) some countable set of
indices depending on Y . Suppose that (Xi )i∈I(Y ) is a sequence of random variables
which are independent conditionally on Y . Let A(Y ) be some event depending on
Y and ∀ j ∈ I(Y ), Bj = Bj (Y, Xj ) be some event depending on Y and Xj . Then,
for any i ∈ I(Y ), and for all sequence of measurable functions (fi )i∈I(Y ) such that
the following quantities exist,




X
X
E [fi (Y, Xi )| Y, Bi ] A#B  ,
E
fi (Y, Xi ) A#B  = E 
i∈I(Y )

i∈I(Y )

E[fi (Y,Xi )1Bi | Y ]
P(Bi | Y )

where E [fi (Y, Xi )| Y, Bi ] =

and A#B = A(Y ) ∩

T


B
.
j
j∈I(Y )

The following lemma is linked to Lemma 4.2.
Lemma B.7. Let N be a linear Hawkes process with no pastR before time 0 (i.e.
t−
N
) = g(t) + 0 h(t − x)N (dx),
N− = ∅) and intensity on (0, ∞) given by λ(t, Ft−
where g and h are as in Proposition B.3 and let for any x, s ≥ 0


Z x

 Lg,h (x) = E
h(x − z)N (dz) Ex,s (N )
s

0


 Gg,h (x) = P (E

x,s (N )) ,

s

Then, for any x, s ≥ 0,
Lg,h
s (x) =

x

Z

 h,h
h (z) + Lh,h
s (z) Gs (z)g(x − z) dz,

(B.18)

s∧x

and
log(Gg,h
s (x))

Z

(x−s)∨0

=

Gh,h
s (x

Z
− z)g(z)dz −

0

x

g(z)dz.

(B.19)

0

h,h
1
∞
In particular, (Lh,h
and is a solution of (4.11)-(4.12).
s , Gs ) is in L × L

Proof. The statement only depends on the distribution of N . Hence, thanks
to

V
Proposition B.4, it is sufficient to consider N = Nanc ∪
∪
V
+
N
.
c
P V ∈Nanc

Let us show (B.18). First, let us write Lg,h
s (x) = E
X∈N h(x − X) Ex,s (N ) .
and note that Lg,h
s (x) = 0 if x ≤ s. The following decomposition holds




X
X

h(x − V ) +
Lg,h
h(x − V − W ) Ex,s (N ) .
s (x) = E
V ∈Nanc

W ∈NcV

According to Lemma B.6 and the following decomposition,
!
Ex,s (N ) = Ex,s (Nanc ) ∩

\
V ∈Nanc

Ex−V,s (NcV

) ,

(B.20)

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let us denote Y = Nanc , XV = NcV and BV = Ex−V,s (NcV ) for all V ∈ Nanc . Let
us fix V ∈ Nanc and compute the conditional expectation of the inner sum with
respect to the filtration of Nanc which is


#
"
X
X


h((x − V ) − W ) Ex−V,s (Nc )
E
h(x − V − W ) Y, BV = E
W ∈Nc

W ∈NcV

=

Lh,h
s (x

− V ),

(B.21)

since, conditionally on Nanc , NcV has the same distribution as N
c which is a linear
R t−
Nc
Hawkes process with conditional intensity λ(t, Ft− ) = h(t) + 0 h(t − z)Nc (dz).
Using the conditional independence of the cluster processes with respect to Nanc ,
one can apply Lemma B.6 and deduce that
"
#
X

g,h
h,h
Ls (x) = E
h(x − V ) + Ls (x − V ) Ex,s (N )
V ∈Nanc

The following argument is inspired by Moller 28 . For every V ∈ Nanc , we say that V
has mark 0 if V has no descendant or himself in [x − s, x) and mark 1 otherwise. Let
0
1
0
us denote Nanc
the set of points with
mark 0 and Nanc
= Nanc \ Nanc
. For any V ∈

h,h
0
Nanc , we have P V ∈ Nanc Nanc = Gs (x−V )1[x−s,x)c (V ), and all the marks are
1
0
are independent Poisson
and Nanc
chosen independently given Nanc . Hence, Nanc
0
h,h
processes and the intensity
of
N
is
given
by
λ(v)
=
g(v)G
anc
s (x − v)1[x−s,x)c (v).
 1
Moreover, the event Nanc = ∅ can be identified to Ex,s (N )and


X

1


Lg,h
h(x − V ) + Lh,h
s (x) = E
s (x − V ) Nanc = ∅
0
V ∈Nanc

Z
=

x−


h,h
h (x − w) + Lh,h
s (x − w) g(w)Gs (x − w)1[x−s,x)c (w)dw

−∞
(x−s)∨0

Z

 h,h
h (x − w) + Lh,h
s (x − w) Gs (x − w)g(w) dw,

=
0

where we used the independence between the two Poisson processes. It suffices to
substitute w by z = x − w in the integral to get the desired formula. Since Gh,h
is
s
h,h
1
bounded, it is obvious that Ls is L .
Then, let us show (B.19). First note that if x < 0, Gg,h
s (x) = 1. Next, following
Q
(B.20) one has Gg,h
(x)
=
E
1
1
Ex,s (Nanc )
s
X∈Nanc Ex−X,s (NcX ) . This is also


Y

Gg,h
1Ex−V,s (NcV )  ,
s (x) = E 1Nanc ∩[x−s,x)=∅
V ∈Nanc ∩[x−s,x)c


= E 1Nanc ∩[x−s,x)=∅


Y
V ∈Nanc ∩[x−s,x)c


Gh,h
s (x − V ) ,

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by conditioning with respect to Nanc . Since Nanc ∩ [x − s, x) is independent of
Nanc ∩ [x − s, x)c , this gives
"
!#
Z
Z x
h,h
g,h
g(z)dz)E exp
log(Gs (x − z))Nanc (dz)
.
Gs (x) = exp(−
[x−s,x)c

x−s

Rx

R
h,h
This leads to log(Gg,h
s (x)) = − x−s g(z)dz+ [x−s,x)c (Gs (x−z)−1)g(z)dz, thanks
to Campbell’s Theorem 23 . Then, (B.19) clearly follows from the facts that if z >
x > 0 then Gh,h
s (x − z) = 1 and g(z) = 0 as soon as z < 0.
Proof of Lemma 4.2 In turn, we use a Banach fixed point argument to prove
that for all s ≥ 0 there exists a unique couple (Ls , Gs ) ∈ L1 (R+ )×L∞ (R+ ) solution
to these equations. To do so, let us first study
Equation (4.11) and define TG,s:

R
Rx
(x−s)∨0
L∞ (R+ ) → L∞ (R+ ) by TG,s (f )(x) := exp 0
f (x − z)h(z)dz − 0 h(z)dz .
The right-hand side is well-defined since h ∈ L1 and f ∈ L∞ . Moreover we have
R

Rx
(x−s)∨0
R (x−s)∨0
kf kL∞
h(z)dz−
h(z)dz
(kf kL∞ −1)
h(z)dz
0
0
0
TG,s (f )(x) ≤ e
≤e
.
This shows that TG,s maps the ball of radius 1 of L∞ into itself, and more precisely
into the intersection of the positive cone and the ball. We distinguish two cases:
Rx
− If x < s, then TG,s (f )(x) = exp(− h(z)dz) for any f , thus, the unique fixed
0

Rx

point is given by Gs : x 7→ exp(− h(z)dz), which does not depend on s > x.
0

− And if x > s, the functional TG,s is a k−contraction in {f ∈ L∞ (R+ ), kf kL∞ ≤
R∞
1}, with k ≤ h(z)dz < 1, by convexity of the exponential. More precisely, using
0

that for all x, y, |ex − ey | ≤ emax(x,y) |x − y| we end up with, for kf k, kgkL∞ ≤ 1,
−

TG,s (f )(x) − TG,s (g)(x) ≤ e

Rx
0

x−s

R

h(z)dz

h(z)dz

e0
Z

≤ kf − gkL∞

Z
kf − gkL∞

(x−s)

h(z)dz
0

h(z)dz.
R+

Hence there exists only one fixed point Gs that we can identify with Gh,h
given
s
in Proposition B.7 and Gh,h
being
a
probability,
G
takes
values
in
[0,
1].
s
s
1
1
Analogously,
we
define
the
functional
T
:
L
(R
L,s
+ ) → L (R+ ) by TL,s (f )(x) :=
Rx
(h (z) + f (z)) Gs (z)h(x − z) dz, and it is easy to check that TL,s is well-defined
s∧x
as well. We similarly distinguish the two cases:
− If x < s, then the unique fixed point is given by Ls (x) = 0.
R∞
− And if x > s, thus TL,s is a k−contraction with k ≤ h(y)dy < 1 in L1 ((s, ∞))
0

since kGs kL∞ ≤ 1 :

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kTL,s (f ) − TL,s (g)kL1 =

R∞ Rx
s


f (z) − g(z) Gs (z)h(x − z)dz dx

s

≤ kGs kL∞

R∞ R∞

f (z) − g(z) h(x − z)dxdz

s v

= kGs kL∞ kf − gkL1 ((s,∞))

R∞

h(y)dy.

0

In the same way, there exists only one fixed point Ls = Lh,h
given by Proposition
s
B.7. In particular Ls (x ≤ s) ≡ 0.
Finally, as a consequence of Equation
R (4.12) we find that if Ls is the unique
fixed point of TL,s , then kLs kL1 (R+ ) ≤

(

∞

0

1−

bounded in L1 with respect to s.

h(y) dy)2

R∞
0

h(y) dy

and therefore Ls is uniformly

Lemma B.8. Let N be a linear Hawkes process withR past before time 0 given by
t−
N
N− and intensity on (0, ∞) given by λ(t, Ft−
) = µ + −∞ h(t − x)N (dx), where µ
is a positive real number and h is a non-negative function with support in R+ , such
1
,exp
that ||h||L1 < 1. If the distribution of N− satisfies (B.15) then (ALλ,loc
) is satisfied.


N
)
. By iProposition B.4, λ(t) =
λ(t)
=
E
λ(t,
F
Proof.
For
all
t
>
0,
let
t−
h
i
hR
R t−
t−
E µ + 0 h(t − x)N>0 (dx) + E −∞ h(t − x)N≤0 (dx) which is possibly infinite.
Let us apply Proposition B.7 with g ≡ µ and s = 0, the choice s = 0 implying
that Et,0 (N>0 ) is of probability 1. Therefore
t−


Z
E µ+




Z t
h(t − x)N>0 (dx) = µ 1 +
(h(x) + L0 (x))dx ,

0

0

where (L0 , G0 = 1) is the solution
of Lemma 4.2 fori s = 0, by identification of
h
R t−
Proposition B.7. Hence E µ + 0 h(t − x)N>0 (dx) ≤ µ(1 + ||h||L1 + ||L0 ||L1 ).
On the other hand, thanks to Lemma B.9, we have
Z

t−

E
−∞





Z t
X 
h(t − x)N≤0 (dx) = E 
h(t − T ) +
[h(t − x) + L0 (t − x)] h(x − T )dx  .
0

T ∈N−

Since all the quantities are non negative, one can exchange all the integrals and
deduce that
Z

t−


h(t − x)N≤0 (dx) ≤ M (1 + ||h||L1 + ||L0 ||L1 ),

E
−∞

with M = supt≥0 E

hR
0
−∞

i
h(t − x)N− (dx) which is finite by assumption. Hence,
1

,exp
λ(t) ≤ (µ + M )(1 + ||h||L1 + ||L0 ||L1 ), and therefore (ALλ,loc
) is satisfied.

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Proof of Proposition B.5 First, by Proposition B.4


N
E λ(t, Ft−
) St− ≥ s =


Z t−
Z t−
h(t − z)N≤0 (dz) Et,s (N )
h(t − z)N>0 (dz) Et,s (N ) + E
µ+E
Z
= µ+E

0
t−


Z
h(t − z)N>0 (dz) Et,s (N>0 ) +E

−∞
t−


h(t − z)N≤0 (dz) Et,s (N≤0 )

−∞

0



h
N
By Lemma B.7, we obtain E λ(t, Ft−
) St− ≥ s = µ + Lµ,h
s (t) + Φ−,P0 (t, s). Idenand Gs = Gh,h
tifying by Lemma 4.2, Ls = Lh,h
s , we obtain
s


h
N
E λ(t, Ft−
) St− ≥ s = Φµ,h
+ (t, s) + Φ−,P0 (t, s).


N
Hence Φµ,h
P0 (t, s) = E λ(t, Ft− ) St− ≥ s .
Lemma B.8 ensures that the assumptions of Theorem 3.3 are fulfilled. Let u
and ρµ,h
3.3. With respect
to the
P0 = ρλ,P0 be defined accordingly as in Theorem


N
)1{St− ≥s} . The first
PDE system, there are two possibilities to express E λ(t, Ft−
h
i
one involves ρλ,P0 and is E ρµ,h
(t,
S
)1
, whereas the second one involves
t−
S
≥s
t−
P0
µ,h
Φµ,h
P0 and is ΦP0 (t, s)P (St− ≥ s) .
R +∞
R +∞
µ,h
This leads to s ρµ,h
u(t, dx), since u(t, ds) is
P0 (t, x)u(t, dx) = ΦP0 (t, s) s
R +∞
the distribution of St− . Let us denote v(t, s) = s u(t, dx): this relation, together
with Equation (3.10) for u,
immediately gives us that v satisfies Equation (4.14)
R +∞
with Φ = Φµ,h
.
Moreover,
u(t, dx) = 1, which gives us the boundary condition
P0
0
in (4.15).

B.3.4. Study of the general case for Φh−,P0 in Proposition B.5
Lemma
B.9. Let consider h a non-negative function with support in R+ such that
R
h < 1, N− a point
hR process on R− with distributioniP0 and N≤0 defined by (B.11).
t−
h
If Φ−,P0 (t, s) := E −∞ h(t − z)N≤0 (dz) Et,s (N≤0 ) , for all s, t ≥ 0, then,


X
h
Φ−,P0 (t, s) = E 
(h(t − T ) + Ks (t, T )) Et,s (N≤0 ) ,
(B.22)
T ∈N−

where Ks (t, u) is given by (4.13).
Proof. Following the decomposition given in Proposition B.4, one has

X
Φh−,P0 (t, s) = E 
h(t − T )
T ∈N−

!!
+

X
V ∈N1T

h(t − V ) +

X
W ∈NcT ,V

h(t − V − W )


Et,s (N≤0 ) ,

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where Et,s (N≤0 ) = Et,s (N− )

T


T 0 ∈N−

Et,s (N1T )

T

V 0 ∈N1T


0
Et−V 0 ,s (NcV ) . Let us fix

T ∈ N− , V ∈ N1T and compute the conditional expectation of the inner sum with
respect to N− and N1T . In the same way as for (B.21) we end up with


X
E
h(t − V − W ) N− , N1T , Et−V,s (NcT,V ) = Lh,h
s (t − V ),
W ∈NcT ,V

since, conditionally on N− and N1T , NcT,V has the same distribution as Nc . Using
the conditional independence of the cluster processes (NcT,V )V ∈N1T with respect


to N− , (N1T )T ∈N− , one can apply Lemma B.6 with Y = N− , (N1T )T ∈N− and
X(T,V ) = NcT,V and deduce that




X
X

h(t − T ) +
 Et,s (N≤0 ) .
Φh−,P0 (t, s) = E 
h(t − V ) + Lh,h
s (t − V )
T ∈N−

V ∈N1T

Let us fix T ∈ N− and compute the conditional expectation of the inner sum with
respect to N− which is


X

T 
Γ := E 
h(t − V ) + Lh,h
(B.23)
s (t − V ) N− , At,s ,
V ∈N1T

where ATt,s = Et,s (N1T ) ∩

T

V 0 ∈N1T


0
Et−V 0 ,s (NcT,V ) . For every V ∈ N1T , we say that

V has mark 0 if V has no descendant or himself in [t − s, t) and mark 1 otherwise.
T,1
T,0
T
Let us denote N1T,0 theset of points with
 mark 0 and N1 = N1 \ N1 .
For any V ∈ N1T , P V ∈ N1T,0 N1T

= Gh,h
s (t−V )1[t−s,t)c (V ) and all the marks

are chosen independently given N1T . Hence, N1T,0 and N1T,1 are independent Poisson
h,h
processes and the intensity of N1T,0 is given
n by λ(v)o= h(v − T )1[0,∞) (v)Gs (t −
v)1[t−s,t)c (v). Moreover, ATt,s is the event N1T,1 = ∅ , so


n
o
X

T,1
h(t − V ) + Lh,h
=∅ 
Γ = E
s (t − V ) N− , N1
V ∈N1T ,0

Z

t−

=



h,h
h(t − v) + Lh,h
s (t − v) h(v − T )1[0,∞) (v)Gs (t − v)1[t−s,t)c (v)dv

−∞

= Ks (t, T ).
Using the independence
of the cluster processes, one can apply Lemma B.6 with

T
Y = N− and XT = N1 , (NcT,V )V ∈N1T and (B.22) clearly follows.
Lemma B.10. Under the assumptions and notations of Proposition B.5 and
Lemma 4.2, the function Φh−,P0 of Proposition B.5 can be identified with (4.18)
under (A1N− ) and with (4.19) under (A2N− ) and (B.15) is satisfied in those two
cases.

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Proof. Using
i
hP
E
(N
)
.
Lemma B.9, we have Φh−,P0 (t, s) = E
(h(t
−
T
)
+
K
(t,
T
))
t,s
≤0
s
T ∈N−


Under A1N− . On the one hand, for every t ≥ 0,
Z

0


h(t − x)N− (dx) = E [h(t − T0 )]

E
−∞

Z

0

Z
h(t − t0 )f0 (t0 )dt0 ≤ ||f0 ||L∞

=
−∞

∞

h(y)dy,
0

hence P0 satisfies (B.15). On
to one point T0 ,
 the other hand, since N− is reduced

1
h
Φ−,P0 (t, s) = P E (N ) E (h(t − T0 ) + Ks (t, T0 )) 1Et,s (N≤0 ) , using the definition
( t,s ≤0 )
of the conditional expectation. First, we compute P(Et,s (N≤0 |T0 ). To do so, we use

T
T0 ,V
T0 Et−V,s (Nc
)
the decomposition Et,s (N≤0 ) = {T0 < t − s} ∩ Et,s (N1T0 ) ∩
V ∈N
1

and the fact that, conditionally on N1T0 , for all V ∈ N1T0 , NcT0 ,V has the same
distribution as Nc to deduce that


h
i
Y




Gs (t − V ) T0  ,
E 1Et,s (N≤0 ) T0 = 1T0 <t−s E 1Et,s (N T0 ) T0 E 
1

T

V ∈N1 0 ∩[t−s,t)c

because the event Et,s (N1T0 ) involves N1T0 ∩ [t − s, t) whereas the product involves
N1T0 ∩ [t − s, t)c , both of those processes being two independent Poisson processes.
Their respective intensities are λ(x) = h(x − T0 )1[(t−s)∨0,t) (x) and λ(x) = h(x −
T0 )1[0,(t−s)∨0) (x), so we end up with
 h
i
 R

t

T0
E
1
T
=
exp
−
h(x
−
T
)1
(x)dx

0
0
[0,∞)

t−s
  Et,s (N1 )
 R

Q


 = exp − (t−s)∨0 [1 − Gs (t − x)] h(x − T0 )dx .
E
G
(t
−
V
)
T

s
0

0

T
V ∈N 0 ∩[t−s,t)c
1

The product of these two last quantities is exactly q(t, s, T0 ) given by (4.13). Note
that q(t, s, T0 ) is exactly the probability that T0 has no descendant in [t − s, t) given
R 0∧(t−s)
T0 . Hence, P (Et,s (N≤0 )) = −∞
q(t, s, t0 )f0 (t0 )dt0 and (4.18) clearly follows.


2
Under AN− . On the one hand, for any t ≥ 0,
Z

0

E
−∞


Z
h(t − x)N− (dx) = E

0

−∞


Z
h(t − x)αdx ≤ α

∞

h(y)dy,

0

hence P0 satisfies (B.15). On the other hand, since we are dealing with a Poisson
process, we can use the same argumentation of marked Poisson processes as in the
proof of Lemma B.7. For every T ∈ N− , we say that T has mark 0 if T has no
0
descendant or himself in [t − s, t) and mark 1 otherwise. Let us denote N−
the set
1
0
of points with mark 0 and N− = N− \ N− . For any T ∈ N− , we have

0
P T ∈ N−
N− = q(t, s, T )1[t−s,t)c (T ),

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0
1
and all the marks are chosen independently given N− . Hence, N−
and N−
are
0
independent Poisson processes and the intensity of N− is given by

λ(z) = α1z≤0 q(t, s, z)1[t−s,t)c (z)


1
Moreover, Et,s (N≤0 ) = N−
= ∅ . Hence,


X
1
= ∅
(h(t − T ) + Ks (t, T )) N−
Φh−,P0 (t, s) = E 
0
T ∈N−

1
0
.
and N−
which gives (4.19) thanks to the independence of N−

B.3.5. Proof of Propositions 4.3 and 4.4
Since we already proved Proposition B.5 and Lemma B.10, to obtain Proposi∞
2
tion 4.3 it only remains to prove that Φµ,h
P0 ∈ L (R+ ), to ensure uniqueness of the
solution by Remark 4.1. To do so, it is easy to see that the assumption h ∈ L∞ (R+ )
combined with Lemma 4.2 giving that Gs ∈ [0, 1] and Ls ∈ L1 (R+ ) ensures that
∞
h
Φµ,h
+ , q and Ks are in L (R+ ). In turn, this implies that Φ−,P0 in both (4.18) and
∞
(4.19) is in L (R+ ), which concludes the proof of Proposition 4.3.
Proof of Proposition 4.4 The method of characteristics leads us to rewrite the
solution v of (4.14)–(4.15) by defining f in ≡ v in on R+ , f in ≡ 1 on R− such that
Rt

f in (s − t)e− (t−s)∨0 Φ(y,s−t+y) dy , when s ≥ t
Rs
v(t, s) =
(B.24)
−
Φ(y+t−s,y) dy
 in
f (s − t)e (s−t)∨0
, when t ≥ s.


1
Let PM
0 be the distribution of the past given by AN− and T0 ∼ U([−M −1, −M ]).
By Proposition 4.3, let vM be the solution of System (4.14)–(4.15) with Φ = Φµ,h
PM
0

in
and v in = vM
, (i.e. the survival function of a uniform variable on [−M − 1, −M ]).
∞
Let also vM be the solution of System (4.14)–(4.15) with Φ = Φµ,h
and v in ≡ 1,
PM
0
and v∞ the solution of (4.21)-(4.22). Then
∞
∞
kvM − v ∞ kL∞ ((0,T )×(0,S)) ≤ kvM − vM
kL∞ ((0,T )×(0,S)) + kvM
− v ∞ kL∞ ((0,T )×(0,S)) .
in
in
By definition of vM
, it is clear that vM
(s) = 1 for s ≤ M, so that Formula (B.24) im∞
∞
plies that vM (t, s) = vM (t, s) as soon as s−t ≤ M and so kvM −vM
kL∞ ((0,T )×(0,S)) =
0 as soon as M ≥ S.
∞
To evaluate the distance kvM
− v ∞ kL∞ ((0,T )×(0,S)) , it remains to prove that
Rt h
−

Φ

M

(y,s−t+y) dy

e 0 −,P0
→ 1 uniformly on (0, T ) × (0, S) for any T > 0, S > 0. For
this, it suffices to prove that Φh−,PM (t, s) → 0 uniformly on (0, T ) × (0, S). Since q
0
given by (4.13) takes values in [exp(−2||h||L1 ), 1], (4.18) implies
R 0∧(t−s)
(h(t − t0 ) + Ks (t, t0 )) 1[−M −1,−M ] (t0 )dt0
h
Φ−,PM (t, s) ≤ −∞R 0∧(t−s)
.
0
exp(−2||h||L1 )1[−M −1,−M ] (t0 )dt0
−∞

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Since ||Gs ||L∞ ≤ 1, Ls and h are non-negative, it is clear that
Z +∞
[h(t − x) + Ls (t − x)] h(x − t0 )dx,
Ks (t, t0 ) ≤
0

and so
Z −M

Z

+∞

Ks (t, t0 )dt0 ≤
−M −1

Z
[h(t − x) + Ls (t − x)]

h(x − t0 )dt0

dx

−M −1

0

Z

!

−M

∞

≤

Z

+∞

[h(t − x) + Ls (t − x)] dx

h(y)dy
ZM∞

0

h(y)dy [||h||L1 + ||Ls ||L1 ] .

≤
M

R∞
Hence, for M large enough

Φh−,PM (t, s)
0

≤

M

h(y)dy [||h||L1 +||Ls ||L1 ]
exp(−2||h||L1 )

→ 0, uniformly

1

in (t, s) since Ls is uniformly bounded in L , which concludes the proof.
B.4. Thinning
The demonstration of Ogata’s thinning algorithm uses a generalization of point
processes, namely the marked point processes. However, only the basic properties
of simple and marked point processes are needed (see 3 for a good overview of point
processes theory). Here (Ft )t>0 denotes a general filtration such that FtN ⊂ Ft for
all t > 0, and not necessarily the natural one, i.e. (FtN )t>0 .
Theorem B.11. Let Π be a (Ft )-Poisson process with intensity 1 on R2+ . Let
1
λ(t, Ft− ) be a non-negative (F
R t )-predictable process which is Lloc a.s. and define the
point process N by N (C) = C×R+ 1[0,λ(t,Ft− )] (z) Π (dt × dz) , for all C ∈ B (R+ ).
Then N admits λ(t, Ft− ) as a (Ft )-predictable intensity. Moreover, if λ is in fact

N
N
), then N admits λ(t, Ft−
) as a FtN FtN -predictable, i.e. λ(t, Ft− ) = λ(t, Ft−
predictable intensity.
Proof. The goal is to apply the martingale characterization of the intensity (Chapter II, Theorem 9 in 3 ). We cannot consider Π as a point process on R+ marked in
R+ (in particular, the point with the smallest abscissa cannot be defined). However,
for every k ∈ N, we can define RΠ(k) , the restriction of Π to the points with ordinate
smaller than k, by Π(k) (C) = C Π (dt × dz) for all C ∈ B (R+ × [0, k]). Then Π(k)
can be seen as a point process on R+ marked in Ek := [0, k] with intensity kernel
1.dz with respect to (Ft ). In the same way, we define N (k) by
Z
N (k) (C) =
1z∈[0,λ(t,Ft− )] Π(k) (dt × dz) for all C ∈ B (R+ ) .
C×R+

Let P(Ft ) be the predictable σ-algebra (see page 8 of 3 ).
Let us denote Ek = B ([0, k]) and P̃k (Ft ) = P (Ft ) ⊗ Ek the associated marked
predictable σ-algebra.

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For any fixed z in E, {(u, ω) ∈ R+ × Ω such that λ(u, Fu− ) (ω) ≥ z} ∈ P (Ft )
since λ is predictable. If Γk = {(u, ω, z) ∈ R+ × Ω × Ek , λ(u, Fu− ) (ω) ≥ z}, then



1
Γk = ∩ ∗ ∪ {(u, ω) ∈ R+ × Ω, λ(u, Fu− ) (ω) ≥ q} × 0, q +
∩ Ek .
n∈N q∈Q+
n
So, Γk ∈ P̃k (Ft ) and 1z∈[0,λ(u,Fu− )]∩Ek is P˜k (Ft )-measurable. Hence, one can apply
the Integration Theorem (Chapter VIII, Corollary 4 in 3 ). So,
Z t Z

1z∈[0,λ(u,Fu− )] M̄ (k) (du × dz)
(Xt )t≥0 :=
is a (Ft )-local martingale
0

where M̄

(k)

Ek

t≥0
(k)

(du × dz) = Π

(du × dz) − dzdu. In fact,
Z t
(k)
Xt = Nt −
min (λ(u, Fu− ), k) du.
0

Rt

(k)
Nt

Yet,
(respectively 0 min (λ(u, Fu− ), k) du) is non-decreasingly converging
Rt
towards Nt (resp. 0 λ(u, Fu− )du). Both of the limits are finite a.s. thanks to the
local integrability of the
(see page 
27 of 3 ). Thanks to monotone conver intensity
Rt
gence we deduce that Nt − 0 λ(u, Fu− )du
is a (Ft )-local martingale. Then,
t≥0

thanks to the martingale characterization of the intensity, Nt admits λ(t, Ft− ) as
an (Ft )-intensity. The last point of the Theorem
 is a reduction of the filtration.
N
), it is a fortiori FtN -progressive and the desired result
Since λ(t, Ft− ) = λ(t, Ft−
follows (see page 27 in 3 ).
This final result can be found in 4 .
Proposition B.12 (Inversion Theorem).

Let N = {Tn }n>0 be a non explosive point process on R+ with FtN -predictable
N
intensity λt = λ(t, Ft−
). Let {Un }n>0 be a sequence of i.i.d. random variables with
N
uniform distribution on [0, 1]. Moreover, suppose that they are independent of F∞
.
Denote Gt = σ (Un , Tn ≤ t). Let N̂ be an homogeneous Poisson process with intensity 1 on R2+ independent of F∞ ∨ G∞ . Define a point process N̄ on R2+ by
Z
Z
X

N̄ ((a, b] × A) =
1(a,b] (Tn ) 1A Un λ(Tn , FTNn − ) +
N̂ (dt × dz)
(a,b]

n>0

N )]
A−[0,λ(t,Ft−

for every 0 ≤ a < b and A ⊂ R+ .
Then, N̄ is an homogeneous
Poisson process
on R2+ with intensity 1 with respect


to the filtration (Ht )t≥0 = Ft ∨ Gt ∨ FtN̂
.
t≥0

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