Marion Legrand

Guillaume Dauphin

Etienne Prévost

Expected effects of a dam removal or mitigation on the persistence of the Atlantic salmon population of the Allier river

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

administrative, economic and social differences among the EU state members. While consolidated states as those in north and west Europe are performing advanced restoration practices, newly incorporated countries in east Europe show differences and difficulties to develop restoration programs. NGOs, farmer Unions and local authorities have important stake on restoration. The first group is made of conservation groups at local, national and European scales with direct interest on the restoration of degraded ecosystems and to provide habitat for species of interest. The second is a key actor interested on enhancing productive uses of the land which should integrate the restoration of degraded natural ecosystems after a number of potential actions delineated by the European Common Agricultural Politics. The third, local authorities, have strong potential for restoration actions derived from their authority to regulate land cover and land uses. So, major power capacity to stimulate and regulate restoration in newly EU members is based on the European Policies regulating productive uses and biodiversity conservation. It is foreseen and expected that the EU Government, Council and Parliament will continue figuring prominently in the power relationships for developing restoration at continental scale while economic-social integration of European human populations takes place.

SY59.1

The dam removal in the Sélune River (France): a long-term interdisciplinary program from the particle to the watershed scale S. Fraisse, J.-M. Roussel, J.-L. Baglinière INRA ESE, Rennes, France Dam removal constitutes a major action to restore connectivity in rivers. However, the risks/benefits balance of such operations remains poorly documented, and long-term studies are lacking. In France, a 16-year scientific program has been initiated to monitor the removal of two dams in the Sélune River (flowing into the Bay of Mont Saint-Michel, UNESCO's world heritage site), and planned in 2019. The ultimate aim of the project is to characterize the nature and the dynamic of ecological changes due to dam removal, and to identify the underlying mechanisms. Involving 20 labs from complementary research fields, the scale of studies conducted in this program ranges from the particle to the watershed. Several studies are currently running according to four interconnected tasks. The first task relates on aquatic biota and its goal is to characterize the evolution of diadromous and resident fish, invertebrates, crayfish, and vegetal (including macro-and microalgae) communities after dam removal. It aims to evaluate the changes in water quality, identify successful evolutionary strategies for diadromous fishes, and highlight fundamental changes in the functioning of freshwater ecosystem after the ocean-river reconnection. The second task focuses on physical river dynamics and aims at characterizing changes in physico-chemical, fine sediments and water fluxes, and resulting impact on geomorphology and aquatic habitats. The third task deals with landscape ecology and intends to monitor changes of riparian vegetation and interactions with agricultural landscape. Finally, the fourth task relies on social geography and aims to study the appropriation process of dam removal by inhabitants of the valley, evaluate the success criteria of this large-scale restoration program and to assess the changes in current ecosystem services. This program constitutes a great opportunity to develop cutting edge technologies such as DIDSON camera, LIDAR full wavelength and optic fibers. The Allier catchment is the last large European river populated with Atlantic salmon. This population is therefore a unique natural heritage. The Poutès dam located in the upstream section of the Allier is one of the major historical reasons for the decline of the Allier population. In 1986, the dam was equipped with a fish lift allowing once again, adult Atlantic salmon to move upstream of the dam. While this has been a significant improvement, Poutès dam is still a major obstacle for the adult upstream migration as well as the juvenile downstream migration. In order to mitigate the negative impacts of the dam, a new development project is being considered. In 2012, a full life cycle hierarchical Bayesian model was developed to describe the population dynamics of the Atlantic salmon population in the Allier catchment. It uses various abundance data sets collected over the last 30 years. This model is used to make projections of the population's future for the next 20 years under 3 different scenarios, all assuming that no stocking would occur, to assess the potential impact of the Poutès dam development project:

(1) no improvement of the downstream and upstream migration at Poutès, (2) 50% improvement of the downstream and upstream migration at Poutès, (3) 100% improvement of the downstream and upstream migration at Poutès (equivalent to dam removal). Our preliminary results indicate that by improving fish passage at only one strategic dam, the situation of the Allier salmon population can be improved significantly, even if this may not suffice to ensure the population persistence on the long term due to other adverse factors. This study was funded by ONEMA, INRA, Loire Grands Migrateurs, Fonds Européens de développement Régional, AELB, Fédération National pour la Pêche en France, Région Centre and Etablissement Public Loire.

SY59.3

Foreseeing reversibility of big dam regulation effects: riparian diversity patterns across seminatural and emerging ecosystems P. Rodríguez-González, A. Albuquerque, T. Ferreira

ISA-UTL, Lisbon, Portugal

Human actions such as big dam-building have markedly transformed landscape generating habitats with novel characteristics and functioning. These emerging ecosystems display both new potential threats and opportunities for the management and conservation of biodiversity. Alqueva dam (SW Iberian Peninsula) created a large impoundment in Guadiana river and an archipelago of permanent islands developed across the reservoir area. In the present study, we aimed at characterizing the biophysical features and floristic composition of Alqueva islands, interpreting the environmental drivers of different functional traits and diversity patterns, and assessing the effect of emerging habitats creation in richness and functional diversity changes over time. We used GIS and ground truth for the biophysical characterization and floristic surveys conducted from 2001-2012, to characterize the composition and structure of plant communities on a representative sample of 64 islands. A functional approach was used by analysing traits abundances across floristic inventories and by calculating a functional diversity index. To assess change in community diversity and structure, we analysed specific and functional diversity, species and trait composition by means of univariate and multivariate data analysis. Our results showed the development stage of emerging habitat colonization in the short and medium term after a reservoir creation. Community trait composition revealed shifts in dispersal traits associated to habitat and landscape change. The newly created water surface favoured hydrochory dispersal traits. Other traits like physiognomic plant type, seed weight and type of leaf were associated to islands physiography. Functional diversity showed a consistent temporal trend change, increasing in the short term, and stabilizing or decreasing in the medium term, across islands despite their different biophysical and floristic features. Understanding regulation effects on riparian functional diversity may add important information when planning regulated habitats management and restoration, since it may contribute considerably to foreseeing the ecosystem resilience against prospective disturbances.

SY59.4

Modeling vegetation colonization in the context of dam removal using remote sensing and in situ observations M. Laslier

SY59. 2

 2 Expected effects of a dam removal or mitigation on the persistence of the Atlantic salmon population of the Allier river M. Legrand 1 , G. Dauphin 2 , E. Prevost 3 1 LOGRAMI, Saint-Pourcin sur Sioule, France, 2 Department of Fisheries and Oceans Northwest Atlantic Fisheries Center, St John's, Canada, 3 INRA ECOBIOP, St-Pée sur Nivelle, France


est destinée au dépôt

Understanding CO2 production and transfer during cheese ripening

Filippo Acerbi

To cite this version:

Filippo Acerbi. Understanding CO2 production and transfer during cheese ripening. Journée de l'Ecole Doctorale Sciences des Procédés, Sciences des Aliments, Jun 2015, Montpellier, France. hal-01837472

Filippo ACERBI

Ingénierie des Agro-Polymères et Technologies Emergentes

Understanding CO2 production and transfer during cheese ripening CO2 is an end product of numerous metabolic pathways of the cheese microflora taking place during ripening. This gaseous compound can affect the final quality of most cheeses, depending on its ability to diffuse and dissolve in both their water and fat phase. CO2 is considered as the driving force for eye growth in cheeses with propionic acid fermentation and its production is fundamental to create space for the growth of moulds in blue cheeses. Furthermore, the loss of CO2 from the cheese to the atmosphere during ripening may be limited by wrapping the cheese in a barrier packaging (foil ripening). Despite the great importance of this molecule during cheese ripening, its production rate, solubility and diffusivity have scarcely been investigated in literature.

We experimentally assessed CO2 solubility, diffusivity and production rate coefficients in model semi-hard cheeses in relation to the main composition and ripening variables: salt in moisture, temperature and cheese age, amongst others.

The permeability of an industrial foil was also assessed at different temperatures of industrial relevance.

All studied composition and ripening variables were found to significantly affect the CO2 production and transfer coefficients, but in a different way. For example increasing temperature was found to linearly decrease CO2 solubility in the cheese, while the CO2 production linearly increased and CO2 diffusion was described with a quadratic function.

The assessed coefficients will be used as input parameters of a bi-dimensional model with the goal of describing the CO2 gradients occurring in a blind semi-hard cheese during industrial ripening.


R Yvinec

Piecewise Deterministic Markov Processes, applications in Biology

Introduction

We deal here with mathematical models used in biology that involved deterministic continuous dynamical systems (mostly: system of Ordinary Differential Equations) perturbed by random discrete events in time. Such perturbation may either take the form of a discontinuous jump, or a change in the rule of the continuous motion (or both). This large class of non diffusion stochastic models has been introduced in the literature and rigorously defined (from the probabilistic perspective) by [Davis, 1984, Davis, 1993] (see also the more recent work by [Jacobsen, 2006]). Applications to biology date back at least to [START_REF] Lasota | The statistical dynamics of recurrent biological events[END_REF]. We emphasize that in this latter work a rather deterministic approach is taken (semigroup theory and evolution equations on densities). Note also that in a more applied literature, such models are usually referred as stochastic hybrid models [START_REF] Kouretas | Stochastic hybrid modeling of biochemical processes[END_REF].

The aim of this note is to present some examples of Piecewise Deterministic Markov Processes (PDMP) applied to biological modeling, and to give an informal description of the probabilistic objects and techniques used for the study of such models for non-probabilist. We will try to convince the reader to the relative simplicity to define such model. In particular, we will detail the pathwise definition of PDMP, widely used for stochastic simulation of trajectories of the process. We will also give informal martingale properties and generatordefinition of PDMP, as those are very useful both for concise definition of PDMP and for mathematical study (in particular long time behavior). We will finally highlight the link with semigroup techniques and evolution equations on densities, which should be more familiar for the reader used to partial differential equations.

The examples taken from the literature include cell-cycle model (or growthfragmentation), bacteria movement (or random intermittent search), ion channels in neuron models (Gating model), integrate and fire models for excitable cells, gene expression models, microtubules growth and active transport inside cells. Most of the examples are taken from [Bressloff, 2014, Malrieu, 2014, Lasota et al., 1992]. An important class of models, that contains almost all examples, concern biochemical reaction network models. Although traditionally represented as system of ordinary differential equations, stochastic version have been recently widely used in systems biology, in order to take into account the observed experimental variability in cellular and molecular biology. We will explain how PDMP then naturally arise as (computational and theoretical) simplification of stochastic chemical reaction network.

Theoretical foundations: Poisson processes

The basic elements of a PDMP is a time-continuous deterministic system and a discrete random process. For the latter, we need a mathematical framework that allows to describe the time repartition of discrete events and to "count" them.

Let pΩ, F , Pq be a probability space with the sample space a non-empty set, F a σ-algebra of subsets of Ω , and P a probability measure on F . Definitions and properties in this section are taken from standard textbooks, see for instance [Bremaud, 1981, Jacobsen, 2006[START_REF] Björk | Introduction to point process from a martingale approach[END_REF]].

The Poisson process

We start by the definition of a simple Point process, and its associated counting process.

Definition 1 (Point process). A point process is a sequence T " pT n q ně1 of R `-valued random variables defined on pΩ, F , Pq such that

• P 0 ă T 1 ď T 2 ď . . . ( " 1 , • P T n ă T n`1 , T n ă 8 ( " P T n ă 8 ( , for all n ě 1 , • P lim nÑ8 T n " 8 ( " 1 .
Thus, a point process is an almost surely increasing sequence of strictly positive, possibly infinite random variables, strictly increasing as long as they are finite and with almost sure limit 8. The interpretation of T n is that, if finite, it is the timepoint at which the nth recording of an event takes place with less than n events occurring altogether (on the time axis R `) if T n " 8. By definition, no event can happen at time 0.

Definition 2 (Counting process). A random process N " tN t ; t P R `u is a counting process if it satisfies the following conditions.

• The trajectories of N are, with probability one, right continuous and piecewise constant.

• The process starts at zero, N 0 " 0.

• For each t, ∆N t " 0 or ∆N t " 1 with probability one. Here ∆N t denotes the jump size of N at time t, or more formally ∆N t " N t ´Nt

´.

In more pedestrian terms, the process N starts at 0 and stays at the level 0 until some random time T 1 when it jumps to N T1 " 1. It then stays at level 1 until another random time T 2 ą T 1 when it jumps to the value N T2 " 2 etc. We will refer to the random times T " pT n q ně1 as the jump (or event) times of N . It should be clear at this point that T is a point process, and that both N and T carry the same information and are related by

N t " ÿ ně1 1 Tnďt .
For instance, ´Nt ě n ¯ô ´Tn ď t ¯.

Remark 1. A third view point (not considered here) is to define a discrete random measure. Although less intuitive, this is the right way to generalize it to counting process (measure) on arbitrary space (useful for individual based modeling for instance).

We now turn to the definition of a Poisson process. There are three equivalent definitions that we now give. The understanding of all three definitions is important to get a "feeling" of what is a Poisson process.

xdef:infy Definition 3 (The infinitesimal definition). A counting process N " tN t ; t P R `u is a Poisson process (of rate λ ą 0) if:

• The process has independent increments, i.e., the number of events in disjoint intervals are independent random variables @ t 0 ă t 1 ă . . . ă t n , ´Nt k ´Nt k´1 ¯1ďkďn are independent.

• The process has stationary increments, i.e.

@ t, h ą 0, the law of ´Nt`h ´Nt ¯is independent of t .

• Only one event can occur at a time. More precisely, there is λ ą 0 such that P N h " 1 ( " λh `ophq, P N h ą 1 ( " ophq .

xdef:axy Definition 4 (The axiomatic definition). A counting process N " tN t ; t P R `u is a Poisson process (of rate λ ą 0) if:

• The process has independent increments, i.e., the number of events in disjoint intervals are independent random variables @ t 0 ă t 1 ă . . . ă t n , ´Nt k ´Nt k´1 ¯1ďkďn are independent.

• The number of events in any time interval of length h is Poisson distributed with mean λh, i.e., for all t, h ą 0:

P N t`h ´Nt " k ( " e ´λh pλhq k k!
xdef:consy Definition 5 (The constructive definition). A counting process N " tN t ; t P R `u is a Poisson process (of rate λ ą 0) if:

N t " ÿ ně1 1 Tnďt ,
for a sequence pT n q having independent identically distributed increments ∆T n " T n ´Tn´1 , with an exponential distribution of parameter λ, i.e. for all n ě 1

P ∆T n ą t ( " e ´λt .
Equivalence of Definition. Def 3 ñ Def 4: One way to prove that N t " P opλtq is by the probability generating function: The rest follows by the independence and stationarity of the increments. Specifically, we show by induction that

G Nt pzq " ÿ kě0 P N t " k ( z k . Since N t`h " N t `pN t`h ´Nt q, by independent increments, G N t`h " G Nt G N t`h ´Nt . Then dG Nt dt " lim hÑ0 G N t`h ´GNt h " lim hÑ0 G Nt " G N t`
P ∆T n ą x | ∆T 1 " x 1 , ∆T 2 " x 2 , . . . ∆T n´1 " x n´1 ( " e ´λx .
To prove that, we use that we can rewrite the event t∆T 1 " x 1 , ∆T 2 " x 2 , . . . ∆T n´1 " x n´1 u " tT 1 " t 1 , T 2 " t 2 , . . . T n´1 " t n´1 u with t i " ř i k"1 x k , and the latter may be re-written as increments of N . Def 5 ñ Def 3 : it only remains to show the independence of the increments. By the lack of memory property of the exponential, we can prove that Zptq " ´TNt`1 ´t, T Nt`2 , T Nt`3 , . . . ¯is independent of ´Nt , T 1 , . . . , T Nt ¯and distributed as Zp0q " ´Tn ¯ně1

. Now, since pN t`h ´Nt q tě0 " HpZptqq for some measurable function H, and all t, this implies the independence of N t and pN t`h ´Nt q tě0 .

The main crucial property of the Poisson process may be summarized in this formula

P N t`h ´Nt " 1 | pN s q sďt ( " P N t`h ´Nt " 1 ( " λhe ´λh .
As the jumps of N are 0 or 1, this may formally written as

E " dN t | pN s q sďt ‰ " λdt
Hence λ is the expected number of jumps per unit of time. Let F t the σ-algebra generated by pN s q sďt , and define

M t " N t ´λt .
We have the Martingale property Proposition 1. For all s, t,

E " M t`s | F t ‰ " M t .
Proof. The proof is trivial using the independence of the increments and the law of the increments, we have

E " N t`s ´Nt | F t ‰ " E " N t`s ´Nt ‰ " λs .
We finish by stating some standard properties of Poisson Processes Proposition 2.

• If N 1 ,N 2 are independent Poisson processes of rate resp. λ 1 and λ 2 , then N defined by N ptq " N 1 ptq `N2 ptq is a Poisson process of intensity λ " λ 1 `λ2 .

• Let N a Poisson process of intensity λ. Suppose that N 1 and N 2 are counting process constructed as follows. For each arrival of N , the first process N 1 is augmented of 1 with probability p, and the second N 2 is augmented of 1 with probability p1´pq, and so independently of the previous arrival. Then N 1 and N 2 are independent Poisson processes of rate resp. pλ and p1 ´pqλ.

• if N a Poisson process of intensity 1, then N λ defined by

N λ ptq " N pλtq ,
is a Poisson process of intensity λ.

• Given that N t " n, the n jump times T 1 , . . . , T n are distributed as the order statistics corresponding to n independent random variables uniformly distributed on p0, tq.

Non homogeneous Poisson process

Again, we state three equivalent definitions ?xdef:inf_inhy? Definition 6 (The infinitesimal definition). A counting process N " tN t ; t P R `u is a non-homogeneous Poisson process of rate function λptq if:

• The process has independent increments.

• For all t, h,

P N t`h ´Nt " 1 ( " λptqh `ophq, P N t`h ´Nt ą 1 ( " ophq .
?xdef:ax_inhy? Definition 7 (The axiomatic definition). A counting process N " tN t ; t P R `u is a non-homogeneous Poisson process of rate function λptq if:

• The process has independent increments.

• The number of events in any time interval pt, t `hs is Poisson distributed with mean ş t`h t λpsqds, e.g., for all t, h ą 0:

P N t`h ´Nt " k ( " e ´pmpt`hq´mptqq pmpt `hq ´mptqq k k! ,
where mptq " ş t 0 λpsqds. ?xdef:cons_inhy? Definition 8 (The constructive definition). A counting process N " tN t ; t P R `u is a non-homogeneous Poisson process of rate function λptq if there exists a standard Poisson process Y (rate " 1) such that:

N t " Y ´ż t 0 λpsqds ¯.
The latter definition deserves a particular comment, as it will be important in the next. A direct calculus shows that Y ´şt`h 0 λpsqds ¯´Y ´şt 0 λpsqds ¯is indeed distributed as a Poisson distribution of mean ş t`h t λpsqds. Conversely, define m ´1ptq " infts ě 0 : mpsq ě tu , t ě 0 .

Then we define Y by Y ptq :" N pm ´1ptqq. Then Y t ´Ys " N pm ´1ptqq´N pm ´1psqq pdq " P o ´mpm ´1ptqq´mpm ´1psqq ¯" P opt´sq .

Note that the non-homogeneous Poisson process satisfy the survival relation (used in many deterministic models, such as renewal population dynamics)

P N t " 0 ( " exp ´´ż t 0 λpsqds ¯.
Furthermore, with

M t " N t ´ż t 0 λpsqds ,
we still have the Martingale property Proposition 3. For all s, t,

E " M t`s | F t ‰ " M t .

Stochastic equation driven by a Poisson process

xssec:stocheqy We are already able at this point to define stochastic equations driven by Poisson processes, which are widely used to simulate stochastic chemical reactions networks. Let Y be a standard Poisson process, and let us consider the equation

N t " N 0 `Y ´ż t 0 λpN s qds ¯, (1) eq
:stocheq

where λ is such that ş t 0 λpN psqqds ă 8. It should be clear that such solution is still a counting process (modulo the initial condition). Hence to prove existence, we only need to know the jump times of N . We will give an explicit construction with the help of the jump times of Y.

Proposition 4. Let pτ n q n be the jump times of Y, then the unique solution of (1) is

N t " ÿ ně1 1 Tnďt ,
where, for all n ě 1 (τ 0 " T 0 " 0)

T n " T n´1 `τn ´τn´1 λpN 0 `n ´1q .

Note that we still have the survival relation, for the law of next jump. At time t, the next jump time T follows

P T ą t `s | F t ( " exp ´´ż t`s t λpN puqqdu ¯" exp ´´λpN ptqqs ¯.
And the Martingale property now reads as follows. With

M t " N t ´ż t 0 λpN s qds ,
we still have the Martingale property Proposition 5. For all s, t,

E " M t`s | F t ‰ " M t .
We are already able to modify such construction to include a continuous dynamical system between jumps, hence to have a constructive definition of a PDMP. We postpone such construction as the generator characterization will also prove to be useful in the next.

Generator

For the homogeneous Poisson process, we have seen that N t ´λt is a Martingale. This is actually a characterization among the counting process (Watanabe theorem). A further even more general characterization among the Markov processes is provided by the generator.

Definition 9 (Generator). For any compactly supported function f : N´ą R `,

we define

Af pnq " λ ´f pn `1q ´f pnq ¯.

We now have ) .

[Hint: for s " 0, use Itô with f pN t q " e iupNtq .] Deduce the Watanabe characterization: N is a Poisson process of intensity λ t .

Proof of proposition 6. We conclude using the fact that M ptq " N ptq ´λt is a Martingale, so that As for the Watanabe characterization, knowing that N ptq satisfies the socalled martingale problem, e.g. equation ( 2) for a sufficiently large class of test functions, will characterize the stochastic process N ptq, among the Markov processes.

f
Note that a differential version of equation ( 2) reads

E " df pN t q | pN s q sďt ‰ " E " Af pN ptqq ‰ dt . And noting E s,n " f pt, N t q ‰ " E " f pt, N t q | N s " n ‰ , we can show that Af pt, nq " lim hÓ0 E t,n " f pt `h, N t`h q ´f pt, N t q ‰ .
We conclude this section by giving the generator of the solution of the first stochastic equation we encountered in subsection 2.3. Proposition 8. The generator of the solution of equation (1) is, for any compactly supported function f , Af pnq " λpnq ´f pn `1q ´f pnq ¯.

Chapman-Kolmogorov equation

A first consequence of the generator is to give any moment equation. Indeed, if A is the generator of N , then, for any test functions

E " f pN ptqq ‰ " E " f pN p0qq ‰ `ż t 0 E " Af pN psqq ‰ ds .
In particular, using f pkq " 1 k"n pkq, we obtain an equation for E " f pN ptqq ‰ " P N ptq " n ( ": ppn, tq. Then, for the solution of equation ( 1), dppn, tq dt " λpn ´1qppn ´1, tq ´λpn, tqppn, tq , for all n ě 0 (with pp´1, tq " 0).

PDMP

We now use our knowledge on stochastic equation driven by Poisson processes to define Piecewise Deterministic Markov processes.

One ode and one Poisson process

We will prove Proposition 9. Assume the ODE

# dX t dt " µpX t q , X 0 " x 0 ,
has a unique global solution for every choice of x 0 , and let β : R Ñ R be an arbitrary chosen function, and N a Poisson process of intensity λ. Then the SDE " dX t " µpX t qdt `βpX t ´qdN t , X 0 " x 0 , (3) eq:sde1

has a unique global solution.

Proof. We have the following concrete algorithm.

• Denote the jump times of N by T 1 , T 2 , . . .

• For every fixed ω, solve the ODE # dX t dt " µpX t q , X 0 " x 0 , on r0, T 1 q. In particular we have determined the value of X T 1 .

• Calculate X T1 by the formula

X T1 " X T 1 `βpX T 1 q .
• Given X T1 " x 1 solve the ODE

# dX t dt " µpX t q , X T1 " x 1 ,
on rT 1 , T 2 q, and so on . . .

We give without proof (similar as proposition 6) the generator of the solution Xptq of equation ( 3):

Af pxq " µpxqf 1 pxq `λ´f px `βpxqq ´f pxq ¯.

We can give a similar construction of solution of SDE when the counting process has now a state-dependent intensity. Using the time change representation, we prove has a unique maximum solution on r0, τ 8 q for a given τ 8 .

Proof. The proof is similar, we simply need to adapt the times of the jumps according to the time change. We have the following concrete algorithm.

• Denote the jump times of Y by T 1 , T 2 , . . .

• For every fixed ω, solve the ODE # dX t dt " µpX t q , X 0 " x 0 , on r0, τ 1 q, where τ 1 is given by τ 1 " inftt ą 0 :

ż t 0 λpX s qds ě T 1 u .
If τ 1 " 8, we are done. Else, we have determined the value of X τ 1 .

• Calculate X τ1 by the formula

X τ1 " X τ 1 `1 . • Given X τ1 " x 1 solve the ODE # dX t dt " µpX t q , X τ1 " x 1 ,
on rτ 1 , τ 2 q, where τ 2 is given by

τ 2 " inftt ą τ 1 : ż τ2 τ1 λpX s qds ě T 2 ´T 1u .
and so on . . . The solution is constructed up to τ 8 " lim n τ n (which may or may not be infinite).

The generator of the solution Xptq of equation ( 4) is given by:

Af pxq " µpxqf 1 pxq `λpxq ´f px `1q ´f pxq ¯.

Remark 3. Note that the first jump time τ 1 in the above proof follows

P τ 1 ě t ( " exp ´´ż t 0 λpX s qds ¯" exp ´´ż xt x0 λpxq µpxq dx ¯.

General PDMP

To define a general PDMP, following [Davis, 1984], we only need to let arbitrary jumps occurring at the discrete event, and to authorize to change the dynamical system as well. This is simply done by adding a discrete variable to the state space, that will keep track of the particular dynamical system we follow, and by choosing a transition measure that will dictate the jump law. More precisely, let I be a countable set, d : I Ñ N and for each i P I, M i an open subset of R dpiq . The state-space E is E " ď iPI M i " tz " px, iq; i P I, x P M i u .

Let pE, Eq the natural Borel space associated to E. The PDMP is determined by the following objects

• Vector fields pH i , i P Iq such that for all i P I, all x 0 P M i , there is a unique global solution in M i of

# dX t dt " H i pX t q , X 0 " x 0 ,
(5) eq:ode_pdmp

• A measurable function λ : E Ñ R `, such that for all z " pi, x 0 q P E, the function t Þ Ñ λpX t q is locally integrable along the solution of equation ( 5)

• A transition measure Q : E ˆE Ñ r0, 1s, such that QpA; zq is a measurable function of z P E for each fixed A P E, and is a probability measure on pE, Eq for each fixed z P E.

A unique global solution is constructed as in the proof of proposition 10. The only needed changes concern the calculus of Z τ1 from Z τ 1 . The post jump location Z τ1 is selected independently of τ 1 , with distribution Qp¨; Z τ 1 q.

The generator of the general PDMP is given by:

Af pi, xq " H i pxq∇ x f pi,

Numerical algorithms

We give below different algorithms that can be useful in various situation, and that are directly related to properties on Poisson process we just saw.

How to simulate several Poisson processes

Suppose one wish to simulate the following system of SDE (any Markovian discrete chemical reaction network)

X i ptq " X i p0q `L ÿ l"1 ν il Y l ´ż t 0 λ l pXpsqqds ¯, (6)
eq:next where pY l q 1ďlďL are independent standard Poisson processes. By the splitting properties of Poisson processes, the law of the solution of equation ( 6) is the same as the solution of

$ ' ' ' & ' ' ' % N ptq " Y ´ż t 0 λpXpsqqds ¯, X i ptq " X i p0q `L ÿ l"1 ν il ż t 0
1 q l´1 pXps ´qq,q l pXps ´qq pξ N ps ´q qdN psq .

(7) eq:gillespi where λpxq " ř L l"1 λ l pxq, q l pxq " ř l i"1 λ i pxq{λpxq, Y is a standard Poisson process, and ξ 0 , ξ 1 , . . . are independent uniform r0, 1s random variable, independent of Y .

Both equivalent formulations equation ( 6) and ( 7) actually suggest two different algorithms, known in the literature as the next reaction method [START_REF] Gibson | Efficient exact stochastic simulation of chemical systems with many species and many channels[END_REF] and the Gillespie [Gillespie, 1977] (or stochastic simulation algorithm, ssa) respectively.

The Gillespie algorithm, corresponding to equations (7) can be described as follows:

• At each step, draw two random uniform r0, 1s independent numbers u, q.

• The next time step is given by ∆T i " ´1{λpXpT i´1 qq logpuq.

• The reaction is chosen according to l " minti ě 1 :

ř i´1 j"1 λ i pxq{λpxq ď ř i j"1 λ i pxq{λpxqu. • Update X accordingly, that is X i " X i `νil .
The next reaction method, corresponding to equations (6) can be described as follows:

• At time 0, for all l " 1..L, draw a random uniform r0, 1s independent number u l , and calculate T l " ´1{λ l pXp0qq logpu l q.

• At each step, let j the index of the smallest T l . Set t " T j .

• Update X accordingly, that is X i " X i `νij .

• For any reaction k ‰ j whose intensities is modified by reaction j, update the new intensity λ 1 k " λ k pXptqq (and keep the old λ k ). Set T k " t pλ k {λ 1 k qpT k ´tq. Forget the old intensity, e.g. λ k " λ 1 k . • For reaction j, draw a random uniform r0, 1s independent number u, and calculate T j " ´1{λ j pXptqq logpuq.

How to simulate an ODE and a Poisson process

Suppose one wish to simulate a hybrid process, given by

$ ' ' & ' ' % N ptq " Y ´ż t 0 λps, Xpsqqds ¯, Xptq " Xp0q `ż t 0 µpXpsqqds `ż t 0 ξ N ps ´qpX ps ´qqdN psq ,
where ξ 0 px 0 q, ξ 1 px 1 q, . . . are are independent random variable, independent of Y , and distributed according to a transition measure Qp¨; xq.

There are basically two options to simulate the jump times of such process. The first one can be convenient if the integral ş t 0 λps, Xpsqqds is known exactly (for instance if it is independent of X):

• At each step, draw a random uniform r0, 1s independent number u.

• Solve the ODE Xptq " Xp0q`ş t 0 µpXpsqqds until the next time step, given by T i`1 " T i `inftt ą 0 : ż Ti`t Ti λps, X s qds ě ´logpuqu • Set t " T i`1 and draw an independent random variable according to Qp¨; Xpt ´qq

• Update X accordingly.

Note that the drawback of this method is that it requires the evaluation of the integral ş λps, X s qds at each step. A second option uses an acceptance/reject method, based on a property of 2d-Poisson process. Assume sup t,x λps, Xpsqq ď λ 0 .

• At each step, draw a random uniform r0, 1s independent number u.

• Set an hypothetical time T 1 i`1 " T i ´1{λ 0 logpuq.

• Solve the ODE Xptq " Xp0q `şt 0 µpXpsqqds on rT i , T 1 i`1 q. • Draw a random uniform r0, 1s independent number q.

• If q ď λpT 1 i`1 , X T 1 i`1 q{λ 0 , set T i`1 " T 1 i`1
, and update X accordingly to the jump law. Else repeat previous step with a new hypothetical jump time T 1 i`1 " T 1 i`1 ´1{λ 0 logpuq (with a new random uniform number u).

General algorithm for a PDMP

General algorithm for a PDMP may be derived from both previous section (exercise!)

Approximations

Approximate algorithm can be drawn from limit theorems (law of large number, central limit theorem), by changing some of the Poisson processes by deterministic terms and/or Brownian term. Other methods such the τ -leaping method choose to fire many reaction in a single step. We refer to [Anderson et al., 2011, Hepp et al., 2014] for more details.

Large time

General results of Markov chain and Markov processes, [Meyn and Tweedie, 1993a, Meyn and Tweedie, 1993b, Meyn, , Costa and Dufour, 2008] Examples on specific models with particular emphasize to the convergence rate [START_REF] Malrieu | Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria[END_REF], Malrieu, 2014, Malrieu, 2012] Examples on specific models using semigroup techniques [START_REF] Tyran-Kamińska | Substochastic semigroups and densities of piecewise deterministic markov processes[END_REF], Pichór et al., 2012, Mackey and Tyran-Kamińska, 2008] 5 Examples from biology

growth-fragmentation

We look at a cell that growth continuously at rate τ pxq, and divides at random times, of intensity λpxq. At division, the size is decreased according to the kernel βpx, yq. We keep track of a single cell. The generator is

Af pxq " τ pxqf 1 pxq `λpxq ż x 0 ´f pyq ´f pxq ¯βpx, dyq .

Particular case: TCP, equal mitosis, cell cycle, integrate and fire.

See [START_REF] Bardet | Total variation estimates for the TCP process[END_REF], Cloez, 2012] for probabilistic approach, and [START_REF] Lasota | The statistical dynamics of recurrent biological events[END_REF][START_REF] Tyran-Kamińska | Substochastic semigroups and densities of piecewise deterministic markov processes[END_REF][START_REF] Doumic | Eigenelements of a general aggregation-fragmentation model[END_REF] for semigroup and entropy approaches.

degrowth-production

xssec:burstiny We look at a quantity (protein concentration, chemical content in a body) that is degraded continuously at rate γpxq, and produced at random times, of intensity λpxq. At production event, a random amount is added according to the kernel κpx, yq. The generator is

Af pxq " ´γpxqf 1 pxq `λpxq ż 8

x ´f pyq ´f pxq ¯κpx, dyq .

Particular case: storage model, bursting in gene expression.

For extensive study of the storage model, see [START_REF] Bardet | Total variation estimates for the TCP process[END_REF], while for more general models applied to gene expression, see [START_REF] Mackey | Dynamic behavior of stochastic gene expression models in the presence of bursting[END_REF]. See also [Cloez, 2012]

Switched flows

The following generic model is widely used

Af px, nq " v n pxqf 1 px, nq `λn pxq ´ÿ k‰n ρ k,n pxqf px, kq ´f px, nq ¯.
with for all px, nq, ř k‰n ρ k,n pxq " 1. Particular case: gene expression, chemotaxis, active transport, random intermittent search, gating model, ion channels, microtubules growth.

We refer to [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF], Malrieu et al., 2012b, Benaïm et al., 2012, Malrieu et al., 2012a, Malrieu, 2012] for generic study of switched flows, and examples of quite surprising results (explosion of a PDMP by switching two stable linear flows).

Biochemical Reaction Network

We consider a network of r 0 chemical reactions involving s 0 chemical species S 1 , . . . , S s0 , represented as

s0 ÿ i"1 ν ik S i Ñ s0 ÿ i"1 ν 1 ik S i , k " 1, . . . , r 0 ,
where the ν ik , ν 1 ik are nonnegative integers (stoichiometric coefficients). Let Xptq P N s0 give the numbers of molecules of each species present at time t. Let ν k " pν ik q 1ďiďs0 be the vector that give the number of molecules consumed in the kth reaction, and ν 1 k " pν 1 ik q 1ďiďs0 the number of molecules produced by the kth reaction. We suppose that each reaction is executed according to a counting process of intensity λ k pxq, that is :sde_crn where Y k are independent standard Poisson processes. Accordingly, we emphasize that the probability that a reaction k fires in a small time interval is given by

Xptq " Xp0q `ÿ k pν 1 k ´νk qY k ´ż t 0 λ k pXpsqqds ¯, (8) eq
P Reaction k occurs in pt, t `δts | F t ( « λ k pXptqqδt .
The function λ k pxq are called reaction rate. A typical example of such function is given by the law of mass-action,

λ k pxq " κ k ź i ˆνik x i ˙,
where κ k ą 0 is the reaction rate constant. The law of large numbers give for the following property for Poisson processes xprop:lgny Proposition 11. let Y be a standard Poisson process, then for each t 0 ą 0,

lim nÑ8 sup tďt0 | 1 n Y pntq ´t |" 0 a.s.
The proposition 11 implies the standard scaling of biochemical reaction network: for large number of species, an appropriate re-scaling of Xptq and of the reaction rate makes the solution of equation ( 8) to converge to the solution of

dx dt " ÿ k κ k pν 1 k ´νk q ź i x ν ik i .
Recent work [START_REF] Kang | Central limit theorems and diffusion approximations for multiscale markov chain models[END_REF], Kurtz and Kang, 2013, Anderson and Kurtz, 2011, Crudu et al., 2012] provides general theorems to reduce equation (8) to a PDMP, when some species are present in large quantity (and/or some reaction occur in a faster time scale). Rather than presenting the general theorems, we will focus on a particular model of gene expression, for which reduced forms have been extensively used.

Standard gene expression models

One of the most simple and widely used model of gene expression consider two variables. The first one is a boolean variable (0 or 1), that represent the state of the gene. The second one is an integer variable, that represents the amount of gene products, to be either mRNA or protein quantity. In the following equation, pY i q i"1,...,4 are independent standard Poisson processes.

$ ' ' & ' ' % X 0 ptq " X 0 p0q `Y1 ´ż t 0 1 tX0psq"0u λ a pX 1 psqqds ¯´Y 2 ´ż t 0 1 tX0psq"1u λ i pX 1 psqqds ¯, X 1 ptq " X 1 p0q `Y3 ´ż t 0 1 tX0psq"1u λ 1 pX 1 psqqds ¯´Y 4 ´ż t 0 γ 1 X 1 psqds ¯. (9) eq:SD1
In such model, the gene state switches between 0 and 1 at rate λ a pX 1 q and λ i pX 1 q. The gene product is degraded at a linear rate γ 1 X 1 . When the gene is "ON" (state 1), the gene product may increase at rate λ 1 pX 1 q. The three functions λ a , λ i , λ 1 represent the auto-regulation of the gene production on its own production. There are constant for constitutive gene (no regulation), and are bounded strictly positive monotonous function in the general case (typical examples are Hill functions).

We refer to [START_REF] Peccoud | Markovian modelling of gene product synthesis[END_REF] for a mathematical study of such model in the case without regulation (λ 1 , λ a , λ i constant). For large amount of gene products (X 1 " 1), this model may be simplified to a switching ODE. The results is limiting model is given by

$ ' ' & ' ' % X 0 ptq " X 0 p0q `Y1 ´ż t 0 1 tX0psq"0u λ a px 1 psqqds ¯´Y 2 ´ż t 0 1 tX0psq"1u λ i px 1 psqqds ¯,
x 1 ptq " x 1 p0q `ż t 0 ´1tX0psq"1u λ 1 px 1 psqq ´γ1 x 1 psq ¯ds.

(10) eq:SC1 The precise result is given by: Theorem 1. Let X N be the solution of equation ( 9) with the rescaled rate functions λ N 1 pxq :" N λ 1 px{N q , γ N 1 :" γ 1 , λ N a pxq :" λ a px{N q , λ N i pxq :" λ i pxq . Assume that λ 1 is a C 1 function of x P R `, and that λ a , λ i , λ 1 are such that equations (10) defines a unique global solution (measurable linearly bounded is sufficient). Then let the rescaled process x N " pX N 0 , x N 1 q defined by x N 1 " X N 1 {N . Assume x N p0q converges in distribution to xp0q, then x N converges in distribution to the PDMP whose law is uniquely defined by equations (10).

A further result in [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] allows to reduce the PDMP defined by equations (10), by "averaging" the discrete motion, assuming it evolves in a fast time scale. The reduced model is know a deterministic ODE, given by dx dt " λ a px 1 q λ a px 1 q `λi px 1 q λ 1 px 1 q ´γ1 x 1 (11) eq:C1

The precise result requires that λ a , λ i , λ 1 are such that the equation ( 11) defines a global flow, not necessarily restrict to evolve in a compact. Furthermore, it requires that the fast motion given by the switch defines an ergodic semigroup, exponentially mixing, and uniformly with respect to the slow variable x 1 . Here, it is easy to see that this semigroup is ergodic, with unique invariant law given by a Bernoulli law of parameter λapx1q λapx1q`λipx1q . Its convergent rate is exponential with rate λ a px 1 q `λi px 1 q. Hence, it is needed to suppose additionally that these rates are bounded with respect to x 1 . As before, we rewrite the limiting theorem given in [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] with our notation: Theorem 2. [Crudu et al., 2012, Theorem 5.1 p. 13] Assume λ 1 P C 1 pR `q and such that the model defined by equation (11) defines a global flow. Assume λ a and λ i are C 1 on R `and bounded, positive such that one of them is strictly positive. Let λ n a " nλ a and λ n i " nλ i with n Ñ 8. Let pX n O ptq, x n 1 ptqq tě0 the stochastic process defined by equations (10), and px 1 ptqq tě0 the solution of equation (11). Assume x n 1 p0q converges in distribution to x 1 p0q in R `, then px n 1 ptqq tě0 converges in distribution to px 1 ptqq tě0 in DpR `; R `q.

The restriction of bounded rate λ a and λ i in [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] is essentially to ensure that the fast dynamics stay in a compact in some sense. Here, because the fast dynamics is on a compact state space, this assumption can be released easily. Note that no convergence on the fast variable is stated here. Convergence do actually occurs, but for a specific topology (in L 8 p0, T q). See [START_REF] Kurtz | Separation of time-scales and model reduction for stochastic reaction networks[END_REF] for general results on averaging methods.

A different limit from the PDMP defined by equations ( 10) may be taken, and was also treated explicitly in [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF]. Now we let λ n i " nλ i and λ n 1 " nλ 1 . Intuitively, the switching variable X n 0 will then spend most of its time in state 0. However, transition from X n 0 " 0 to X n 0 " 1 will still be possible (and will not vanish as n Ñ 8). Convergence of X n 0 to 0 will hold in L 1 p0, tq for any finite finite time t. When X n 0 " 1, production of x 1 is suddenly very high, but for a brief time. Although x 1 follows a deterministic trajectory, the timing of its trajectory is stochastic. At the limit, this drastic production episode becomes a discontinuous jump, of a random size. All happen as the two successive jumps of X 0 (from 0 to 1 and back to 0) coalesce into a single one, and create a discontinuity in x 1 . In such case, convergence cannot hold in the cad-lag space DpR `; R `q with the Skorohod topology. The authors in [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] were able to prove tightness in L p pr0, T s, R `q, 1 ď p ă 8. Their result requires the additional assumption that all rates λ 1 ,λ i and λ a are linearly bounded, and either λ a or λ i is bounded with respect to x 1 . This is needed to get a bound on x 1 in L 8 pr0, T s, R `q. The limiting theorem reads Theorem 3. [Crudu et al., 2012, Theorem 6.1 p. 17] Assume λ 1 P C 1 pR `q and let λ n i " nλ i and λ n 1 " nλ 1 with n Ñ 8. Let pX n O ptq, x n 1 ptqq tě0 the stochastic process defined by equations (10). Assume x n 1 p0q converges in distribution to x 1 p0q in R `, and X n O p0q converges in distribution to 0. The reaction rates λ 1 ,λ i and λ a are such that

• there exists α ą 0 such that λ i px 1 q ě α for all x 1 ;

• there exists M 1 ą 0 such that

λ 1 px 1 q ď M 1 px 1 `1q, λ a px 1 q ď M 1 px 1 `1q, λ i px 1 q ď M 1 px 1 `1q;
• In addition either λ a or λ 1 is bounded with respect to x 1 .

Then pX n 0 ptqq tě0 converges in distribution to 0 in L 1 pr0, T s, t0, 1uq and px n 1 ptqq tě0 converges in distribution to the stochastic process whose generator is given by Aϕpx 1 q " ´γ1 x 1 Bϕ Bx 1

`λa px 1 q ż 8 0 ´ϕpφ 1 pt, x 1 qq ´ϕpx 1 q ¯λi pφ 1 pt, x 1 qqe ´şt 0 λipφ1ps,x1qqds dt, (12) eq:geneswitchCrudu for every ϕ P C 1 b pR `q and where φ 1 pt, x 1 q is the flow associated to 9

x " λ 1 pxq,

xp0q " x 1 .

Remark 4. Note that equation ( 12) may be rewritten as

Af pxq " ´γpxqf 1 pxq `λa pxq ż 8

x ´f pyq ´f pxq ¯hpy, xqdy, (13) eq:geneswitchCrudu2

for every f P C 1 b pR `q and where hpy, xq is given by hpy, xq " ´ν1 pyq νpxq , y ąě x ą 0., νpxq " exp `´ż x 0 λ i pzq λ 1 pzq dz ¯.

Let us now present in detail three examples.

Example 1. Consider the special case where both regulation rates λ 1 and λ i are constant. Then the flow φ 1 is easily calculated and we have φ 1 pt, x 1 q " x 1 `λ1 t, t ě 0, ż t 0 λ i k i pφ 1 ps, x 1 qqds " λ i t, and the generator eq. ( 12) becomes Aϕpx 1 q " ´γ1 x 1 Bϕ Bx 1 `λa px 1 q ż 8 0 ´ϕpx 1 `zq ´ϕpx 1 q ¯λi λ 1 e ´λi λ 1

z dz, which is a "Bursting" model (see subsection 5.2), with an exponential jump size distribution of mean parameter λ1 λi . Such rate has an easy interpretation, being the number of molecules created during an ON period of the gene.

Other choice of regulation rate leads to different model, as illustrated in the next two examples.

Example 2. Let λ 1 be a constant and λ i px 1 q " λ i x 1 `k0 (linear negative regulation), so that φ 1 pt, x 1 q " x 1 `λ1 t, t ě 0, ż t 0 λ i k i pφ 1 ps, x 1 qqds " pλ i x 1 `k0 qt `λ1 λ i 2 t 2 , and the generator eq. ( 12) becomes Aϕpx 1 q " ´γ1 x 1 Bϕ Bx 1 `λa k a px 1 q

ż 8 x1 ´ϕpzq´ϕpx 1 q ¯λi z `k0 λ 1 e ´z´x 1 λ 1 " λi z`x 1 2 `k0 ı dz.
The limiting model is then a bursting model where the jump distribution is a function of the jump position, and has a Gaussian tail.

Example 3. Let λ 1 px 1 q " x 1 and λ i be a constant (positive linear regulation), so that φ 1 pt, x 1 q " x 1 e λ1t , t ě 0, ż t 0 λ i k i pφ 1 ps, x 1 qqds " λ i t,

and the generator eq. ( 12) becomes Aϕpx 1 q " ´γ1 x 1 Bϕ Bx 1 `λa k a px 1 q

ż 8 x1 ´ϕpzq ´ϕpx 1 q ¯λi λ 1 x λ i λ 1 z ´1´λ i λ 1 dz.
This time, the limiting model is a bursting model where the jump distribution is a function of the jump position with a power-law tail.

The model obtained in equation ( 10) is a switching ODE. The advantage of reducing the model compared to equation ( 9) is to be able to calculate explicitly the stationary density (if it exists), which leads to specific strategy to prove convergence in large time towards a stationary state. We state the result for a slightly generalized model: x 1 ptq " x 1 p0q `ż t 0 ´1tX0psq"1u λ 1 px 1 psqq `1tX0psq"0u λ 0 px 1 psqq ¯ds.

(14) eq:SC1_generic Suppose there exists y ˚ă y ˚such that " λ 0 py ˚q " 0 , λ 1 py ˚q ą 0 , λ 0 py ˚q ă 0 , λ 1 py ˚q " 0 .

Then, if x 1 p0q takes values on py ˚, y ˚q, so is x 1 ptq for all t ě 0. Moreover, the stationary density of the continuous variable x 1 of the solution of equation ( 14) is solution of the first order differential equation

B Bx u "
´" λ a λ 0 `λi λ 1 `λ1 0 λ 1 {λ 0 ´λ1 1 λ 0 {λ 1 λ 1 ´λ0 ı u , x P py ˚, y ˚q .

We refer to [START_REF] Mackey | Dynamic behavior of stochastic gene expression models in the presence of bursting[END_REF] for systematic study to the model obtained in equation (13)

  .

Exercise 2 .

 2 Let N the solution of equation (1). Calculate dE " manners. In which case do you obtain closed equations?

Theorem 4 .

 4 Let us look at the equation0u λ a px 1 psqqds ¯´Y 2 ´ż t 0 1 tX0psq"1u λ i px 1 psqqds ¯,

  Nt pzq " e λtpz´1q , which is the p.g.f of a Poisson random variable.

	h ´Nt h G N t`h ´Nt " 1 ´λh `λhz `ophq , dG Nt dt " λpz ´1qG Nt . intervals and use the Poisson limit theorem, e.g. By the definition 3, so that lim nÑ8 Binpn, θ n qptkuq " P opλtqptkuq , if nθ n Ñ λt. Def 4 ñ Def 5: The distribution of the first jump time follows from ´1ı Hence G Remark 2. A different proof consider subdividing r0, ts in n (n Ñ 8) disjoints P T 1 ą t ( " P N t " 0 ( " e ´λt

  The sample path of N (t Þ Ñ N t pwq) are piecewise continuous and have finite variation. As such, we may define a Lebesgue-Stieljes integral of bounded function on finite time integral against the measure pdN t pwqq t . Note that such measure may formally be represented asÿ kě1 δ T k pwq pdsq ,where pT k q is the sequence of jump times associated to the Poisson process. As an important first step, we state the Proposition 7 (Itô Lemma). Let N be a homogeneous Poisson process of rate λ. Then, for any measurable function f ,

	f pN ptqq " f pN p0qq	`ż t	´f pN ps ´q `1q ´f pN ps ´qq ¯dN psq .
						0
	Proof. A simple calculus yields
	f pN ptqq " f pN p0qq	`Nptq ÿ	pf pkq ´f pk ´1qq
						k"1
		" f pN p0qq	`Nptq ÿ	pf pN pT k qq ´f pN pT k´1 qqq
						k"1
		" f pN p0qq	`Nptq ÿ	pf pN pT ḱ `1qq ´f pN pT ḱ qqq
						k"1
		" f pN p0qq	`ż t
	E " e iupNt´Nsq | F s	‰	" exp	!	pmptq ´mpsqqpe iu ´1q
		f pN ptqq ´f pN p0qq	´ż t	Af pN psqqds	(2) eq:gene
						0
	is a F t -Martingale. This holds as well for any function f such that E " f pN ptqq ‰	ă
	8, E " Af pN ptqq ‰	ă 8, for all t ě 0.

xprop:geney Proposition 6. Let N be a homogeneous Poisson process of rate λ. Then, for any compactly supported function f , 0 ´f pN ps ´q `1q ´f pN ps ´qq ¯dN psq .

Exercise 1. Let N a counting process and t Þ Ñ λ t a deterministic process such that

M t " N t ´ż t 0 λ s ds ,

is a F t -Martingale. Shows that for all t ě s,