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# Parametric estimation and LAN property of the birth-death-move process with mutations

Lisa Balsollier, Frédéric Lavancier

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Lisa Balsollier, Frédéric Lavancier. Parametric estimation and LAN property of the birth-death-move process with mutations. 2024. hal-04554652

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# Parametric estimation and LAN property of the birth-death-move process with mutations

Lisa Balsollier, Frédéric Lavancier

► To cite this version:

Lisa Balsollier, Frédéric Lavancier. Parametric estimation and LAN property of the birth-death-move process with mutations. 2024. hal-04554652

HAL Id: hal-04554652

https://hal.science/hal-04554652v1

Preprint submitted on 26 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# Parametric estimation and LAN property of the birth-death-move process with mutations

Lisa Balsollier¹,² and Frédéric Lavancier³

¹LMJL, UMR 6629 CNRS, Nantes Université, lisa.balsollier@cnrs.fr

²SERPICO Project-Team, Centre Inria de l'Université de Rennes

³Univ Rennes, Ensai, CNRS, CREST, frederic.lavancier@ensai.fr

## Abstract

A birth-death-move process with mutations is a Markov model for a system of marked particles in interaction, that move over time, with births and deaths. In addition the mark of each particle may also change, which constitutes a mutation. Assuming a parametric form for this model, we derive its likelihood expression and prove its local asymptotic normality. The efficiency and asymptotic distribution of the maximum likelihood estimator, with an explicit expression of its covariance matrix, is deduced. The underlying technical assumptions are showed to be satisfied by several natural parametric specifications. As an application, we leverage this model to analyse the joint dynamics of two types of proteins in a living cell, that are involved in the exocytosis process. Our approach enables to quantify the so-called colocalization phenomenon, answering an important question in microbiology.

*Keywords:* Likelihood estimation, Jump move process, Colocalization, Microbiology.

## 1 Introduction

We address the parametric inference of birth-death-move processes with mutations, in short BDMM. These processes model the dynamics of a system
# Parametric estimation and LAN property of the birth-death-move process with mutations

Lisa Balsollier¹,² and Frédéric Lavancier³

¹LMJL, UMR 6629 CNRS, Nantes Université, lisa.balsollier@cnrs.fr

²SERPICO Project-Team, Centre Inria de l'Université de Rennes

³Univ Rennes, Ensai, CNRS, CREST, frederic.lavancier@ensai.fr

## Abstract

A birth-death-move process with mutations is a Markov model for a system of marked particles in interaction, that move over time, with births and deaths. In addition the mark of each particle may also change, which constitutes a mutation. Assuming a parametric form for this model, we derive its likelihood expression and prove its local asymptotic normality. The efficiency and asymptotic distribution of the maximum likelihood estimator, with an explicit expression of its covariance matrix, is deduced. The underlying technical assumptions are showed to be satisfied by several natural parametric specifications. As an application, we leverage this model to analyse the joint dynamics of two types of proteins in a living cell, that are involved in the exocytosis process. Our approach enables to quantify the so-called colocalization phenomenon, answering an important question in microbiology.

*Keywords:* Likelihood estimation, Jump move process, Colocalization, Microbiology.

## 1 Introduction

We address the parametric inference of birth-death-move processes with mutations, in short BDMM. These processes model the dynamics of a system
of particles in $\mathbb{R}^d$ that move through time, while some new particles may
appear and existing ones may disappear. In addition, each particle may be
marked by a label, that can change over time, which we call a mutation.
This type of dynamics is observed in epidemiology [21], in ecology [26, 24]
and in microbiology [15, 2]. For example, the application we will consider
later concerns the dynamics of proteins in a living cell, involved in the ex-
ocytosis process and observed near the plasma membrane of the cell: these
proteins move within the cell, and for biological and photochemical reasons,
some of them disappear while others appear. Moreover the motion of each
protein can be of three types, the regime of which constitutes its label, and
this regime may change over time, in line with what we call a mutation.

Formally, a BDMM process is defined through several characteristics: first, the intensity of births, of deaths, and of mutations, that are functions of the particle configurations and rule the waiting time before the next birth (or death, or mutation, respectively); second, the transition kernels that specify how a birth (or a death or a mutation) occurs when it happens; third, a continuous Markov process that drives the motion of the system of particles between two jumps, whether this jump is a birth, a death, or a mutation. Assuming a parametric form for each of these characteristics, we are interested by their inference given a single realisation of the process on a finite time interval. In this contribution, we address this question by maximum likelihood and we provide theoretical guarantees by proving the local asymptotic normality (LAN) property of the model, when the time interval increases, under some regularity conditions. The latter implies the efficiency and asymptotic normality of the maximum likelihood estimator, with an explicit and estimable asymptotic covariance matrix.

Birth-death-move processes (without mutations) have been introduced
in [15] and further studied in [14], with a particular focus on their ergodic
properties. These processes share close similarities with Markovian particle
systems with killing and jumps as studied in [18, 17], branching processes [1]
and spatially structured population models [3]. The introduction of possi-
ble mutations does not change much the probabilistic properties of a birth-
death-move process, as we will justify it when needed in the paper. From a
statistical point of view, non-parametric estimation of their intensity func-
tions has been considered in [15]. When there is no move and no mutation,
the process boils down to a spatial birth-death process, as introduced in [25].
Parametric maximum likelihood inference of a spatial birth-death process
when $d = 1$ has been considered in [22], with an applications to the analy-
of particles in $\mathbb{R}^d$ that move through time, while some new particles may
appear and existing ones may disappear. In addition, each particle may be
marked by a label, that can change over time, which we call a mutation.
This type of dynamics is observed in epidemiology [21], in ecology [26, 24]
and in microbiology [15, 2]. For example, the application we will consider
later concerns the dynamics of proteins in a living cell, involved in the ex-
ocytosis process and observed near the plasma membrane of the cell: these
proteins move within the cell, and for biological and photochemical reasons,
some of them disappear while others appear. Moreover the motion of each
protein can be of three types, the regime of which constitutes its label, and
this regime may change over time, in line with what we call a mutation.

Formally, a BDMM process is defined through several characteristics: first, the intensity of births, of deaths, and of mutations, that are functions of the particle configurations and rule the waiting time before the next birth (or death, or mutation, respectively); second, the transition kernels that specify how a birth (or a death or a mutation) occurs when it happens; third, a continuous Markov process that drives the motion of the system of particles between two jumps, whether this jump is a birth, a death, or a mutation. Assuming a parametric form for each of these characteristics, we are interested by their inference given a single realisation of the process on a finite time interval. In this contribution, we address this question by maximum likelihood and we provide theoretical guarantees by proving the local asymptotic normality (LAN) property of the model, when the time interval increases, under some regularity conditions. The latter implies the efficiency and asymptotic normality of the maximum likelihood estimator, with an explicit and estimable asymptotic covariance matrix.

Birth-death-move processes (without mutations) have been introduced
in [15] and further studied in [14], with a particular focus on their ergodic
properties. These processes share close similarities with Markovian particle
systems with killing and jumps as studied in [18, 17], branching processes [1]
and spatially structured population models [3]. The introduction of possi-
ble mutations does not change much the probabilistic properties of a birth-
death-move process, as we will justify it when needed in the paper. From a
statistical point of view, non-parametric estimation of their intensity func-
tions has been considered in [15]. When there is no move and no mutation,
the process boils down to a spatial birth-death process, as introduced in [25].
Parametric maximum likelihood inference of a spatial birth-death process
when $d = 1$ has been considered in [22], with an applications to the analy-
sis of the displacement of dunes. The same method has been employed in [27] for particular cases of a spatial birth-death process in $d = 2$, in a view to model openings and closures of shops in Tokyo. In these references, no theoretical study is provided and by definition the particles do not move. In [17], the LAN property of the closely related Markovian model of particle systems with killing and jumps is established, under a very general, quite abstract, setting. In our contribution, we specify the likelihood expression of the general birth-death-move process with mutations and we similarly show the LAN property under mild assumptions, that we prove to be satisfied for standard parametric specifications of the model.

The article is organised as followed. In Section 2, the formal definition of the BDMM process is provided, along with some specific examples for each of its characteristics. In Section 3, we derive the likelihood expression of the model and we prove its LAN property. We in particular show that the associated technical hypotheses are satisfied for the natural examples presented in Section 2, under mild assumptions. In Section 4, a simulation study is carried out, showing the performances of the maximum likelihood estimator of some parameters of the model, along with the estimation of the associated confidence region. The setting is close to that of the dataset analysed subsequently, in order to provide certain assurances regarding the reliability of the conclusions concerning it. This dataset is a video sequence showing Langerin proteins and Rab-11 proteins in a living cell, acquired by fluorescence microscopy [4]. As detailed in Section 4.2, their dynamics are consistent with the realisation of BDMM process. Leveraging on this model and its estimation, we prove that Langerin proteins are colocalized with Rab-11 proteins, and we quantify this phenomenon, answering an important biological question. Finally, an appendix gathers the proofs of our theoretical results.

The data, the Python code concerning the simulation study and the code for processing the data are available in our online GitHub repository at https://github.com/balsollier-lisa/Parametric-estimation-of-the-BDMM-process.
sis of the displacement of dunes. The same method has been employed in [27] for particular cases of a spatial birth-death process in $d = 2$, in a view to model openings and closures of shops in Tokyo. In these references, no theoretical study is provided and by definition the particles do not move. In [17], the LAN property of the closely related Markovian model of particle systems with killing and jumps is established, under a very general, quite abstract, setting. In our contribution, we specify the likelihood expression of the general birth-death-move process with mutations and we similarly show the LAN property under mild assumptions, that we prove to be satisfied for standard parametric specifications of the model.

The article is organised as followed. In Section 2, the formal definition of the BDMM process is provided, along with some specific examples for each of its characteristics. In Section 3, we derive the likelihood expression of the model and we prove its LAN property. We in particular show that the associated technical hypotheses are satisfied for the natural examples presented in Section 2, under mild assumptions. In Section 4, a simulation study is carried out, showing the performances of the maximum likelihood estimator of some parameters of the model, along with the estimation of the associated confidence region. The setting is close to that of the dataset analysed subsequently, in order to provide certain assurances regarding the reliability of the conclusions concerning it. This dataset is a video sequence showing Langerin proteins and Rab-11 proteins in a living cell, acquired by fluorescence microscopy [4]. As detailed in Section 4.2, their dynamics are consistent with the realisation of BDMM process. Leveraging on this model and its estimation, we prove that Langerin proteins are colocalized with Rab-11 proteins, and we quantify this phenomenon, answering an important biological question. Finally, an appendix gathers the proofs of our theoretical results.

The data, the Python code concerning the simulation study and the code for processing the data are available in our online GitHub repository at https://github.com/balsollier-lisa/Parametric-estimation-of-the-BDMM-process.
# 2 Definition of the BDMM process

## 2.1 State space

We denote by $(\mathbf{X}_t)_{t \ge 0}$ the BDMM process. At each time $t \ge 0$, $\mathbf{X}_t$ describes the configuration of a set of marked particles. Each particle $x \in \mathbf{X}_t$ reads $x = (z, m)$ where $z \in \Lambda \subset \mathbb{R}^{p+q}$ encodes the spatial location of the particle in $\mathbb{R}^p$, $p \ge 1$, along with its possible continuous mark in $\mathbb{R}^q$, $q \ge 0$, and where $m \in \mathbb{M}$ is a discrete mark (or label), $\mathbb{M}$ being the finite set of possible labels. In the following, we set $d = p + q$, so that $\Lambda \subset \mathbb{R}^d$.

At each time $t$, $\mathbf{X}_t$ represents the set of all alive particles, where the ordering does not matter and the cardinality may change over time. For this reason, for any $n \ge 1$, we introduce the natural projection

$$ \Pi_n : (x_1, \dots, x_n) \in (\Lambda \times \mathbb{M})^n \mapsto \{x_1, \dots, x_n\} \quad (1) $$

that identifies two elements $(x_1, \dots, x_n)$ and $(y_1, \dots, y_n)$ of $(\Lambda \times \mathbb{M})^n$ if there exists a permutation $\pi$ of $\{1, \dots, n\}$ such that $x_i = y_{\pi(i)}$ for any $1 \le i \le n$.

We set for any $n \ge 1$, $E_n = \Pi_n((\Lambda \times \mathbb{M})^n)$. The state space of $\mathbf{X}_t$ for $t \ge 0$ is then

$$ E = \bigsqcup_{n \ge 0} E_n, $$

where $E_0 = \{\emptyset\}$ consists of the empty configuration.

The cardinality of a configuration $\mathbf{x} \in E$ will be denoted by $n(\mathbf{x})$, so that $\mathbf{x} \in E_{n(\mathbf{x})}$, and we write

$$ \mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}, $$

where $z_i \in \Lambda$ and $m_i \in \mathbb{M}$. In the following we will sometimes assimilate $(\Lambda \times \mathbb{M})^n$ with $\Lambda^n \times \mathbb{M}^n$ for convenience, so that we may also write $\mathbf{x} = \{(\mathbf{z}, \mathbf{m})\}$ where $\mathbf{z} = (z_1, \dots, z_{n(\mathbf{x})})$ and $\mathbf{m} = (m_1, \dots, m_{n(\mathbf{x})})$. Similarly, we will write the configuration at time $t$ of the BDMM process in either way

$$ \mathbf{X}_t = \{x_{1,t}, \dots, x_{n,t}\} = \{(z_{1,t}, m_{1,t}) \dots, (z_{n,t}, m_{n,t})\} \text{ or } \mathbf{X}_t = \{(\mathbf{Z}_t, \mathbf{m}_t)\}, $$

where $n = n(\mathbf{X}_t)$. Moreover, for $\mathbf{x} \in E_n$ and $(z, m) \in \Lambda \times \mathbb{M}$, we write $\mathbf{x} \cup (z, m)$ for $\{x_1, \dots, x_n, (z, m)\} \in E_{n+1}$, and for $n \ge 1$ and $1 \le i \le n$, we write $\mathbf{x} \setminus x_i$ for $\{x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n\} \in E_{n-1}$.
# 2 Definition of the BDMM process

## 2.1 State space

We denote by $(\mathbf{X}_t)_{t \ge 0}$ the BDMM process. At each time $t \ge 0$, $\mathbf{X}_t$ describes the configuration of a set of marked particles. Each particle $x \in \mathbf{X}_t$ reads $x = (z, m)$ where $z \in \Lambda \subset \mathbb{R}^{p+q}$ encodes the spatial location of the particle in $\mathbb{R}^p$, $p \ge 1$, along with its possible continuous mark in $\mathbb{R}^q$, $q \ge 0$, and where $m \in \mathbb{M}$ is a discrete mark (or label), $\mathbb{M}$ being the finite set of possible labels. In the following, we set $d = p + q$, so that $\Lambda \subset \mathbb{R}^d$.

At each time $t$, $\mathbf{X}_t$ represents the set of all alive particles, where the ordering does not matter and the cardinality may change over time. For this reason, for any $n \ge 1$, we introduce the natural projection

$$ \Pi_n : (x_1, \dots, x_n) \in (\Lambda \times \mathbb{M})^n \mapsto \{x_1, \dots, x_n\} \quad (1) $$

that identifies two elements $(x_1, \dots, x_n)$ and $(y_1, \dots, y_n)$ of $(\Lambda \times \mathbb{M})^n$ if there exists a permutation $\pi$ of $\{1, \dots, n\}$ such that $x_i = y_{\pi(i)}$ for any $1 \le i \le n$.

We set for any $n \ge 1$, $E_n = \Pi_n((\Lambda \times \mathbb{M})^n)$. The state space of $\mathbf{X}_t$ for $t \ge 0$ is then

$$ E = \bigsqcup_{n \ge 0} E_n, $$

where $E_0 = \{\emptyset\}$ consists of the empty configuration.

The cardinality of a configuration $\mathbf{x} \in E$ will be denoted by $n(\mathbf{x})$, so that $\mathbf{x} \in E_{n(\mathbf{x})}$, and we write

$$ \mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}, $$

where $z_i \in \Lambda$ and $m_i \in \mathbb{M}$. In the following we will sometimes assimilate $(\Lambda \times \mathbb{M})^n$ with $\Lambda^n \times \mathbb{M}^n$ for convenience, so that we may also write $\mathbf{x} = \{(\mathbf{z}, \mathbf{m})\}$ where $\mathbf{z} = (z_1, \dots, z_{n(\mathbf{x})})$ and $\mathbf{m} = (m_1, \dots, m_{n(\mathbf{x})})$. Similarly, we will write the configuration at time $t$ of the BDMM process in either way

$$ \mathbf{X}_t = \{x_{1,t}, \dots, x_{n,t}\} = \{(z_{1,t}, m_{1,t}) \dots, (z_{n,t}, m_{n,t})\} \quad \text{or} \quad \mathbf{X}_t = \{(\mathbf{Z}_t, \mathbf{m}_t)\}, $$

where $n = n(\mathbf{X}_t)$. Moreover, for $\mathbf{x} \in E_n$ and $(z, m) \in \Lambda \times \mathbb{M}$, we write $\mathbf{x} \cup (z, m)$ for $\{x_1, \dots, x_n, (z, m)\} \in E_{n+1}$, and for $n \ge 1$ and $1 \le i \le n$, we write $\mathbf{x} \setminus x_i$ for $\{x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n\} \in E_{n-1}$.
We equip E with the Borel σ-algebra $\mathcal{E}$ and we endow E with the distance $d_1$ defined for $\mathbf{x}$ and $\mathbf{y}$ in E such that $n(\mathbf{x}) \le n(\mathbf{y})$ by

$$d_1(\mathbf{x}, \mathbf{y}) = \frac{1}{n(\mathbf{y})} \left( \min_{\pi \in S_n(\mathbf{y})} \sum_{i=1}^{n(\mathbf{x})} (\|x_i - y_{\pi(i)}\| \wedge 1) + (n(\mathbf{y}) - n(\mathbf{x})) \right),$$

with $d_1(\mathbf{x}, \emptyset) = 1$ and where $S_n$ denotes the set of permutations of $\{1, \dots, n\}$. This distance makes in particular the function $\mathbf{x} \mapsto n(\mathbf{x})$ continuous on E, and further implies some nice topological properties for $(E, d_1)$, see [28] and [14] for details.

## 2.2 Dynamics of a BDMM process

The dynamics of a BDMM process alternates continuous motions and jumps. Inter-jumps motions concern the location and/or the continuous mark of all alive particles, that move in $\Lambda$. Jumps are of three types: either a birth (when a new particle appears), or a death (when an existing particle disappears), or a mutation (when an existing particle changes its label). Accordingly, the dynamics is based on the three following elements:

1. A continuous homogeneous Markov process $(\mathbf{Y}_t)_{t \ge 0}$ on E that drives the motion of all particles of $(\mathbf{X}_t)_{t \ge 0}$ between two jumps. This process will not affect the label of the particles, neither their cardinality. Given an initial condition $\mathbf{Y}_0 = \mathbf{x}$, $(\mathbf{Y}_t)_{t \ge 0}$ will be defined as the solution of a stochastic differential equation, as described and exemplified in Section 2.3.3.

2. Three continuous non-negative functions $\beta, \delta$ and $\tau$ on E, that refers respectively to the birth, death and mutation intensities. These functions govern the waiting times between the jumps of $(\mathbf{X}_t)_{t \ge 0}$. Heuristically, the probability that a birth occurs in the interval $[t, t + dt]$ given that the particles are in the configuration $\mathbf{X}_t$ at time $t$ is $\beta(\mathbf{X}_t)dt$, and similarly for $\delta$ and $\tau$. We will denote $\alpha = \beta + \delta + \tau$ the total jump intensity of the process, that we assume to be bounded.

3. Three transition kernels $K_\beta, K_\delta$, and $K_\tau$ from $E \times \mathcal{E}$ to $[0, 1]$, that specify how a jump occurs when it happens. For example, given that a birth occurs in a configuration $\mathbf{x}$, $K_\beta(\mathbf{x}, A)$ is the probability that the
We equip E with the Borel σ-algebra $\mathcal{E}$ and we endow E with the distance $d_1$ defined for $\mathbf{x}$ and $\mathbf{y}$ in E such that $n(\mathbf{x}) \le n(\mathbf{y})$ by

$$d_1(\mathbf{x}, \mathbf{y}) = \frac{1}{n(\mathbf{y})} \left( \min_{\pi \in S_{n}(\mathbf{y})} \sum_{i=1}^{n(\mathbf{x})} (\|x_i - y_{\pi(i)}\| \wedge 1) + (n(\mathbf{y}) - n(\mathbf{x})) \right),$$

with $d_1(\mathbf{x}, \emptyset) = 1$ and where $S_n$ denotes the set of permutations of $\{1, \dots, n\}$. This distance makes in particular the function $\mathbf{x} \mapsto n(\mathbf{x})$ continuous on E, and further implies some nice topological properties for $(E, d_1)$, see [28] and [14] for details.

## 2.2 Dynamics of a BDMM process

The dynamics of a BDMM process alternates continuous motions and jumps. Inter-jumps motions concern the location and/or the continuous mark of all alive particles, that move in $\Lambda$. Jumps are of three types: either a birth (when a new particle appears), or a death (when an existing particle disappears), or a mutation (when an existing particle changes its label). Accordingly, the dynamics is based on the three following elements:

1. A continuous homogeneous Markov process $(\mathbf{Y}_t)_{t \ge 0}$ on E that drives the motion of all particles of $(\mathbf{X}_t)_{t \ge 0}$ between two jumps. This process will not affect the label of the particles, neither their cardinality. Given an initial condition $\mathbf{Y}_0 = \mathbf{x}$, $(\mathbf{Y}_t)_{t \ge 0}$ will be defined as the solution of a stochastic differential equation, as described and exemplified in Section 2.3.3.

2. Three continuous non-negative functions $\beta, \delta$ and $\tau$ on E, that refers respectively to the birth, death and mutation intensities. These functions govern the waiting times between the jumps of $(\mathbf{X}_t)_{t \ge 0}$. Heuristically, the probability that a birth occurs in the interval $[t, t + dt]$ given that the particles are in the configuration $\mathbf{X}_t$ at time $t$ is $\beta(\mathbf{X}_t)dt$, and similarly for $\delta$ and $\tau$. We will denote $\alpha = \beta + \delta + \tau$ the total jump intensity of the process, that we assume to be bounded.

3. Three transition kernels $K_\beta, K_\delta$, and $K_\tau$ from $E \times \mathcal{E}$ to $[0, 1]$, that specify how a jump occurs when it happens. For example, given that a birth occurs in a configuration $\mathbf{x}$, $K_\beta(\mathbf{x}, A)$ is the probability that the
new configuration $\mathbf{x} \cup (z, m)$ belongs to $A \in \mathcal{E}$, where $(z, m)$ denotes the new particle, and similarly for $K_\delta$ and $K_\tau$.

Details and examples about these three basic characteristics of the process are provided in the next section.

Let us specifically describe the algorithmic definition of the process. This iterative construction follows [15] and [14], and may serve as a simulation procedure, see [15] for details. Let $(\mathbf{Y}_t^{(i)})_{t \ge 0}$, $i \ge 0$, be a sequence of processes on $E$, identically distributed as $(\mathbf{Y}_t)_{t \ge 0}$. Starting from the initial configuration $\mathbf{X}_0$ at time $T_0 = 0$, we iteratively build the process as follows.

i) Given $\mathbf{X}_0$, generate the $n(\mathbf{X}_0)$ continuous trajectories of $(\mathbf{Y}_t^{(0)})_{t \ge 0}$.

ii) Given $\mathbf{X}_0$ and $(\mathbf{Y}_t^{(0)})_{t \ge 0}$, generate the first inter-jump time $T_1 - T_0$ according to the cumulative distribution function

$$F_1(t) = 1 - \exp \left( - \int_0^t \alpha(\mathbf{Y}_u^0) du \right), \quad t > 0. \quad (2)$$

The process until time $T_1$ is given by the generated trajectories, i.e.

$$(\mathbf{X}_t)_{T_0 \le t < T_1} = (\mathbf{Y}_{t-T_0}^0)_{T_0 \le t < T_1}.$$

iii) Given $T_1$ and $\mathbf{X}_{T_1^-} = \mathbf{Y}_{T_1-T_0}^0$, generate the first jump:

* it is a birth with probability $\beta(\mathbf{X}_{T_1^-})/\alpha(\mathbf{X}_{T_1^-})$, in which case $\mathbf{X}_{T_1^-} \sim K_\beta(\mathbf{X}_{T_1^-}, \cdot)$;

* it is a death with probability $\delta(\mathbf{X}_{T_1^-})/\alpha(\mathbf{X}_{T_1^-})$, in which case $\mathbf{X}_{T_1^-} \sim K_\delta(\mathbf{X}_{T_1^-}, \cdot)$;

* it is a mutation with probability $\tau(\mathbf{X}_{T_1^-})/\alpha(\mathbf{X}_{T_1^-})$, in which case $\mathbf{X}_{T_1^-} \sim K_\tau(\mathbf{X}_{T_1^-}, \cdot)$.

iv) Return to step i) with $T_0 \leftarrow T_1$ and $\mathbf{X}_0 \leftarrow \mathbf{X}_{T_1}$ to generate the new trajectories $(\mathbf{Y}_t^{(1)})_{t \ge 0}$ starting from $\mathbf{X}_{T_1}$, the next jumping time $T_2$, and so on.

The sequence of jumping times of the BDMM process $(\mathbf{X}_t)_{t \ge 0}$ is $(T_n)_{n \ge 1}$. For $t > 0$, we denote by $N_t$ the number of jumps on $[0, t]$, i.e.

$$N_t = \sum_{i \ge 1} \mathbf{1}_{T_i \le t}.$$
new configuration $\mathbf{x} \cup (z, m)$ belongs to $A \in \mathcal{E}$, where $(z, m)$ denotes the new particle, and similarly for $K_\delta$ and $K_\tau$.

Details and examples about these three basic characteristics of the process are provided in the next section.

Let us specifically describe the algorithmic definition of the process. This iterative construction follows [15] and [14], and may serve as a simulation procedure, see [15] for details. Let $(\mathbf{Y}_t^{(i)})_{t \ge 0}$, $i \ge 0$, be a sequence of processes on $E$, identically distributed as $(\mathbf{Y}_t)_{t \ge 0}$. Starting from the initial configuration $\mathbf{X}_0$ at time $T_0 = 0$, we iteratively build the process as follows.

i) Given $\mathbf{X}_0$, generate the $n(\mathbf{X}_0)$ continuous trajectories of $(\mathbf{Y}_t^{(0)})_{t \ge 0}$.

ii) Given $\mathbf{X}_0$ and $(\mathbf{Y}_t^{(0)})_{t \ge 0}$, generate the first inter-jump time $T_1 - T_0$ according to the cumulative distribution function

$$F_1(t) = 1 - \exp \left( - \int_0^t \alpha(\mathbf{Y}_u^0) du \right), \quad t > 0. \quad (2)$$

The process until time $T_1$ is given by the generated trajectories, i.e.

$$(\mathbf{X}_t)_{T_0 \le t < T_1} = (\mathbf{Y}_{t-T_0}^0)_{T_0 \le t < T_1}.$$

iii) Given $T_1$ and $\mathbf{X}_{T_1} = \mathbf{Y}_{T_1-T_0}^0$, generate the first jump:

* it is a birth with probability $\beta(\mathbf{X}_{T_1}^-)/\alpha(\mathbf{X}_{T_1}^-)$, in which case $\mathbf{X}_{T_1} \sim K_\beta(\mathbf{X}_{T_1}^-)$;

* it is a death with probability $\delta(\mathbf{X}_{T_1}^-)/\alpha(\mathbf{X}_{T_1}^-)$, in which case $\mathbf{X}_{T_1} \sim K_\delta(\mathbf{X}_{T_1}^-)$;

* it is a mutation with probability $\tau(\mathbf{X}_{T_1}^-)/\alpha(\mathbf{X}_{T_1}^-)$, in which case $\mathbf{X}_{T_1} \sim K_\tau(\mathbf{X}_{T_1}^-)$.

iv) Return to step i) with $T_0 \leftarrow T_1$ and $\mathbf{X}_0 \leftarrow \mathbf{X}_{T_1}$ to generate the new trajectories $(\mathbf{Y}_t^{(1)})_{t \ge 0}$ starting from $\mathbf{X}_{T_1}$, the next jumping time $T_2$, and so on.

The sequence of jumping times of the BDMM process $(\mathbf{X}_t)_{t \ge 0}$ is $(T_n)_{n \ge 1}$. For $t > 0$, we denote by $N_t$ the number of jumps on $[0, t]$, i.e.

$$N_t = \sum_{i \ge 1} \mathbf{1}_{T_i \le t}.$$
Similarly, we denote by $N_t^\beta$, $N_t^\delta$ and $N_t^\tau$ the number of births, deaths and mutations, respectively, before $t$. By our assumption that the total jump intensity $\alpha$ is bounded, we have that $N_t < \infty$ for any $t \ge 0$, i.e. there is no explosion of the process.

The BDMM process, as defined by the above iterative construction, is a particular case of a jump-move process, as studied in [14]. We deduce that it is a homogeneous Markov process with respect to its natural filtration $(\mathcal{F}_t)_{t \ge 0}$. We refer to the latter reference for the expression of the infinitesimal generator of the process and further probabilistic properties. In the following, we will denote by $\mathbb{P}_x$ and $\mathbb{E}_x$ all probabilities and expectations given that $\mathbf{X}_0 = \mathbf{x}$.

## 2.3 Elements of the model and examples

In this section, we describe more precisely each characteristics of the process, that are the intensity functions, the transition kernels, and the move process $(\mathbf{Y}_t)_{t \ge 0}$. We also provide various examples.

### 2.3.1 Intensity functions

The birth, death and mutation intensity functions are such that, given $\mathbf{X}_t = \mathbf{x}$, a birth (resp. a death and a mutation) occurs in $(t, t+h]$ with probability $\beta(\mathbf{x})h+o(h)$ (resp. $\delta(\mathbf{x})h+o(h)$ and $\tau(\mathbf{x})h+o(h)$) as $h \to 0$. These intensities can also be seen in the following way: $\beta(\mathbf{X}_{t-})$ (resp. $\delta(\mathbf{X}_{t-})$ and $\tau(\mathbf{X}_{t-})$) is the intensity of the associated counting process $N_t^\beta$ (resp. $N_t^\delta$ and $N_t^\tau$). These interpretations are consequences of the specific form (2) of the waiting time before the next jump, see [15] for details.

To set some examples, let us denote by $\gamma$ any of $\beta$, $\delta$ or $\tau$. The most simple situation is when the intensity function is a constant rate, that is $\gamma(\mathbf{x}) = \gamma$ for any $\mathbf{x} \in E$, where $\gamma > 0$. Then there is in average $\gamma\Delta$ new events that occur in any time interval of size $\Delta$, whatever the configuration of $\mathbf{X}_t$ is. Another typical setting is when the intensity is proportional to the cardinality, that is $\gamma(\mathbf{x}) = \gamma n(\mathbf{x})$ for $\mathbf{x} \in E$ and $\gamma > 0$. This reflects the situation where each particle has its own rate $\gamma > 0$ and the particles do not interact, so that the total rate over all particles is $\gamma n(\mathbf{x})$. These two examples are observed in the biological applications studied in [15] and [2]. More complicated examples of intensity functions are also considered in [15], where $\gamma(\mathbf{x})$ depends on the underlying Voronoï tessellation induced by $\mathbf{x}$.
Similarly, we denote by $N_t^\beta$, $N_t^\delta$ and $N_t^\tau$ the number of births, deaths and mutations, respectively, before $t$. By our assumption that the total jump intensity $\alpha$ is bounded, we have that $N_t < \infty$ for any $t \ge 0$, i.e. there is no explosion of the process.

The BDMM process, as defined by the above iterative construction, is a particular case of a jump-move process, as studied in [14]. We deduce that it is a homogeneous Markov process with respect to its natural filtration $(\mathcal{F}_t)_{t \ge 0}$. We refer to the latter reference for the expression of the infinitesimal generator of the process and further probabilistic properties. In the following, we will denote by $\mathbb{P}_x$ and $\mathbb{E}_x$ all probabilities and expectations given that $\mathbf{X}_0 = \mathbf{x}$.

## 2.3 Elements of the model and examples

In this section, we describe more precisely each characteristics of the process, that are the intensity functions, the transition kernels, and the move process $(\mathbf{Y}_t)_{t \ge 0}$. We also provide various examples.

### 2.3.1 Intensity functions

The birth, death and mutation intensity functions are such that, given $\mathbf{X}_t = \mathbf{x}$, a birth (resp. a death and a mutation) occurs in $(t, t+h]$ with probability $\beta(\mathbf{x})h+o(h)$ (resp. $\delta(\mathbf{x})h+o(h)$ and $\tau(\mathbf{x})h+o(h)$) as $h \to 0$. These intensities can also be seen in the following way: $\beta(\mathbf{X}_{t-})$ (resp. $\delta(\mathbf{X}_{t-})$ and $\tau(\mathbf{X}_{t-})$) is the intensity of the associated counting process $N_t^\beta$ (resp. $N_t^\delta$ and $N_t^\tau$). These interpretations are consequences of the specific form (2) of the waiting time before the next jump, see [15] for details.

To set some examples, let us denote by $\gamma$ any of $\beta$, $\delta$ or $\tau$. The most simple situation is when the intensity function is a constant rate, that is $\gamma(\mathbf{x}) = \gamma$ for any $\mathbf{x} \in E$, where $\gamma > 0$. Then there is in average $\gamma\Delta$ new events that occur in any time interval of size $\Delta$, whatever the configuration of $\mathbf{X}_t$ is. Another typical setting is when the intensity is proportional to the cardinality, that is $\gamma(\mathbf{x}) = \gamma n(\mathbf{x})$ for $\mathbf{x} \in E$ and $\gamma > 0$. This reflects the situation where each particle has its own rate $\gamma > 0$ and the particles do not interact, so that the total rate over all particles is $\gamma n(\mathbf{x})$. These two examples are observed in the biological applications studied in [15] and [2]. More complicated examples of intensity functions are also considered in [15], where $\gamma(\mathbf{x})$ depends on the underlying Voronoï tessellation induced by $\mathbf{x}$.
### 2.3.2 Transition kernels

Denoting $\gamma$ for any of $\beta, \delta$ or $\tau$, we assume that for any $\mathbf{x}$ the transition kernel $K_{\gamma}(\mathbf{x}, \cdot)$ admits a density $k_{\gamma}(\mathbf{x}, \mathbf{y})$, $\mathbf{y} \in E$, with respect to a measure $\nu_{\gamma}(\mathbf{x}, \cdot)$, that is for all $\mathbf{x} \in E$ and $\mathbf{y} \in E$,

$$K_{\gamma}(\mathbf{x}, d\mathbf{y}) = k_{\gamma}(\mathbf{x}, \mathbf{y})\nu_{\gamma}(\mathbf{x}, d\mathbf{y}).$$

We now specify $k_{\gamma}$ and $\nu_{\gamma}$ in each case, whether the transition concerns a birth, a death or a mutation.

For a birth transition, we assume that for any $F \in \mathcal{E}$, $\nu_{\beta}(\mathbf{x}, F) = 0$ except if there exist $A \subset \Lambda$ and $\mathcal{I} \subset \mathbb{M}$ such that

$$F = \mathbf{x} \cup (A \times \mathcal{I}) := \{\mathbf{y} \in E, \exists (z, m) \in A \times \mathcal{I}, \mathbf{y} = \mathbf{x} \cup (z, m)\}.$$

In this case we set $\nu_{\beta}(\mathbf{x}, \mathbf{x} \cup (A \times \mathcal{I})) = |A| \times |\mathcal{I}|$ to be the Lebesgue measure on $\Lambda \times \mathbb{M}$. This choice of $\nu_{\beta}$ ensures that the birth transition produces the addition of exactly one particle to the configuration $\mathbf{x}$, as expected for a birth. For the density $k_{\beta}(\mathbf{x}, \mathbf{y})$, we assume that

$$k_{\beta}(\mathbf{x}, \mathbf{y}) = \begin{cases} p_m k_{\beta}^m(\mathbf{x}, z) & \text{if } \exists (z, m) \in \Lambda \times \mathbb{M}, \mathbf{y} = \mathbf{x} \cup (z, m), \\ 0 & \text{otherwise,} \end{cases} \quad (3)$$

where $p_m \in [0, 1]$ is the probability that the new particle has the label $m \in \mathbb{M}$, with $\sum_{m \in \mathbb{M}} p_m = 1$, and $k_{\beta}^m(\mathbf{x}, \cdot)$ is the density for the location of the new particle in $\Lambda$ given that its label is $m$.

A simple example of birth transition is the uniform law for the label and the location, which corresponds to $p_m = 1/|\mathbb{M}|$ for all $m \in \mathbb{M}$ and $k_{\beta}^m(\mathbf{x}, z) = 1/|\Lambda|$, for all $z \in \Lambda$ and $m \in \mathbb{M}$. More sophisticated examples are provided below.

**Example 1 (Birth kernel as a mixture of normal laws).** In this example, a new particle is more likely to appear close to existing particles. The birth density is a mixture of isotropic normal distributions, centred at each existing particle, with deviation $\sigma > 0$. Specifically, for any configuration $\mathbf{x} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}$ and any $m \in \mathbb{M}$,

$$k_{\beta}^m(\mathbf{x}, z) = \frac{1}{n(\mathbf{x})} \sum_{i=1}^{n(\mathbf{x})} \frac{1}{v(z_i, z)} \exp \left( - \frac{\|z - z_i\|^2}{2\sigma^2} \right),$$
### 2.3.2 Transition kernels

Denoting $\gamma$ for any of $\beta, \delta$ or $\tau$, we assume that for any $\mathbf{x}$ the transition kernel $K_{\gamma}(\mathbf{x}, \cdot)$ admits a density $k_{\gamma}(\mathbf{x}, \mathbf{y})$, $\mathbf{y} \in E$, with respect to a measure $\nu_{\gamma}(\mathbf{x}, \cdot)$, that is for all $\mathbf{x} \in E$ and $\mathbf{y} \in E$,

$$K_{\gamma}(\mathbf{x}, d\mathbf{y}) = k_{\gamma}(\mathbf{x}, \mathbf{y})\nu_{\gamma}(\mathbf{x}, d\mathbf{y}).$$

We now specify $k_{\gamma}$ and $\nu_{\gamma}$ in each case, whether the transition concerns a birth, a death or a mutation.

For a birth transition, we assume that for any $F \in \mathcal{E}$, $\nu_{\beta}(\mathbf{x}, F) = 0$ except if there exist $A \subset \Lambda$ and $\mathcal{I} \subset \mathbb{M}$ such that

$$F = \mathbf{x} \cup (A \times \mathcal{I}) := \{\mathbf{y} \in E, \exists (z, m) \in A \times \mathcal{I}, \mathbf{y} = \mathbf{x} \cup (z, m)\}.$$

In this case we set $\nu_{\beta}(\mathbf{x}, \mathbf{x} \cup (A \times \mathcal{I})) = |A| \times |\mathcal{I}|$ to be the Lebesgue measure on $\Lambda \times \mathbb{M}$. This choice of $\nu_{\beta}$ ensures that the birth transition produces the addition of exactly one particle to the configuration $\mathbf{x}$, as expected for a birth. For the density $k_{\beta}(\mathbf{x}, \mathbf{y})$, we assume that

$$k_{\beta}(\mathbf{x}, \mathbf{y}) = \begin{cases} p_m k_{\beta}^m(\mathbf{x}, z) & \text{if } \exists (z, m) \in \Lambda \times \mathbb{M}, \mathbf{y} = \mathbf{x} \cup (z, m), \\ 0 & \text{otherwise,} \end{cases} \quad (3)$$

where $p_m \in [0, 1]$ is the probability that the new particle has the label $m \in \mathbb{M}$, with $\sum_{m \in \mathbb{M}} p_m = 1$, and $k_{\beta}^m(\mathbf{x}, \cdot)$ is the density for the location of the new particle in $\Lambda$ given that its label is $m$.

A simple example of birth transition is the uniform law for the label and the location, which corresponds to $p_m = 1/|\mathbb{M}|$ for all $m \in \mathbb{M}$ and $k_{\beta}^m(\mathbf{x}, z) = 1/|\Lambda|$, for all $z \in \Lambda$ and $m \in \mathbb{M}$. More sophisticated examples are provided below.

**Example 1 (Birth kernel as a mixture of normal laws).** In this example, a new particle is more likely to appear close to existing particles. The birth density is a mixture of isotropic normal distributions, centred at each existing particle, with deviation $\sigma > 0$. Specifically, for any configuration $\mathbf{x} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}$ and any $m \in \mathbb{M}$,

$$k_{\beta}^m(\mathbf{x}, z) = \frac{1}{n(\mathbf{x})} \sum_{i=1}^{n(\mathbf{x})} \frac{1}{v(z_i, z)} \exp \left( - \frac{\|z - z_i\|^2}{2\sigma^2} \right),$$
where $v(z_i, z) = \int_{\Lambda} \exp\left(-\frac{\|z-z_i\|^2}{2\sigma^2}\right) dz$. We may easily extend this example by requiring that a particle with label $m$ can only appear close to particles with labels $m'$.

**Example 2 (Birth kernel driven by a potential).** In this example the birth kernel is given through a function $V: E \to \mathbb{R}$ by:

$$k_{\beta}^{m}(\mathbf{x}, z) = \frac{1}{a^{m}(\mathbf{x})} e^{-(V(\mathbf{x} \cup (z, m)) - V(\mathbf{x}))}$$

where $a^m(\mathbf{x}) = \int_{\Lambda} e^{-(V(\mathbf{x} \cup (z, m)) - V(\mathbf{x}))} dz$. Given a configuration $\mathbf{x}$, a new particle is more likely to appear in the vicinity of points $z \in \Lambda$ that make $V(\mathbf{x} \cup (z, m)) - V(\mathbf{x})$ minimal. Following the statistical physics terminology, the function $V$ can then be seen as a potential that the system tends to minimizing at each birth. A typical instance, for $\mathbf{x} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}$, is $V(\mathbf{x}) = \sum_{i \neq j} \Phi_{m_i, m_j}(z_i - z_j)$ for some pair potential functions $\Phi_{m, m'}$, $m, m' \in \mathbb{M}$, see [14] for some examples.

Concerning the mutation transition kernel, we set for any $\mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\}$, where $x_i = (z_i, m_i)$, and any $F \in \mathcal{E}$,

$$\nu_{\tau}(\mathbf{x}, F) = \sum_{i=1}^{n(\mathbf{x})} \sum_{m \in \mathbb{M}} \mathbf{1}_{(\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m) \in F},$$

so that a transition only concerns a change of label of an existing particle. For the density $k_{\tau}$, we assume that

$$k_{\tau}(\mathbf{x}, \mathbf{y}) = \begin{cases} s(x_i, \mathbf{x})q_m(x_i, \mathbf{x}) & \text{if } \exists x_i \in \mathbf{x}, \exists m \in \mathbb{M}, \mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m), \\ 0 & \text{otherwise.} \end{cases}$$

In this expression, $s(x_i, \mathbf{x}) \in [0, 1]$ is the probability that the particle $x_i$ in $\mathbf{x}$ changes his label and $q_m(x_i, \mathbf{x})$ is the probability that, given that the particle $x_i \in \mathbf{x}$ mutates, its label changes from $m_i$ to $m$. We thus have $\sum_{i=1}^{n(\mathbf{x})} s(x_i, \mathbf{x}) = 1$ and $\sum_{m \in \mathbb{M}} q_m(x_i, \mathbf{x}) = 1$ with $q_{m_i}(x_i, \mathbf{x}) = 0$. An example is provided below.

**Example 3 (Mutation kernel given by a transition matrix).** A natural example is when the particle to modify is uniformly chosen among
where $v(z_i, z) = \int_{\Lambda} \exp\left(-\frac{\|z-z_i\|^2}{2\sigma^2}\right) dz$. We may easily extend this example by requiring that a particle with label $m$ can only appear close to particles with labels $m'$.

**Example 2 (Birth kernel driven by a potential).** In this example the birth kernel is given through a function $V: E \to \mathbb{R}$ by:

$$k_{\beta}^{m}(\mathbf{x}, z) = \frac{1}{a^{m}(\mathbf{x})} e^{-(V(\mathbf{x} \cup (z, m)) - V(\mathbf{x}))}$$

where $a^m(\mathbf{x}) = \int_{\Lambda} e^{-(V(\mathbf{x} \cup (z, m)) - V(\mathbf{x}))} dz$. Given a configuration $\mathbf{x}$, a new particle is more likely to appear in the vicinity of points $z \in \Lambda$ that make $V(\mathbf{x} \cup (z, m)) - V(\mathbf{x})$ minimal. Following the statistical physics terminology, the function $V$ can then be seen as a potential that the system tends to minimizing at each birth. A typical instance, for $\mathbf{x} = \{(z_1, m_1), \dots, (z_{n(\mathbf{x})}, m_{n(\mathbf{x})})\}$, is $V(\mathbf{x}) = \sum_{i \neq j} \Phi_{m_i, m_j}(z_i - z_j)$ for some pair potential functions $\Phi_{m, m'}$, $m, m' \in \mathbb{M}$, see [14] for some examples.

Concerning the mutation transition kernel, we set for any $\mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\}$, where $x_i = (z_i, m_i)$, and any $F \in \mathcal{E}$,

$$\nu_{\tau}(\mathbf{x}, F) = \sum_{i=1}^{n(\mathbf{x})} \sum_{m \in \mathbb{M}} \mathbf{1}_{(\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m) \in F},$$

so that a transition only concerns a change of label of an existing particle. For the density $k_{\tau}$, we assume that

$$k_{\tau}(\mathbf{x}, \mathbf{y}) = \begin{cases} s(x_i, \mathbf{x})q_m(x_i, \mathbf{x}) & \text{if } \exists x_i \in \mathbf{x}, \exists m \in \mathbb{M}, \mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m), \\ 0 & \text{otherwise.} \end{cases}$$

In this expression, $s(x_i, \mathbf{x}) \in [0, 1]$ is the probability that the particle $x_i$ in $\mathbf{x}$ changes his label and $q_m(x_i, \mathbf{x})$ is the probability that, given that the particle $x_i \in \mathbf{x}$ mutates, its label changes from $m_i$ to $m$. We thus have $\sum_{i=1}^{n(\mathbf{x})} s(x_i, \mathbf{x}) = 1$ and $\sum_{m \in \mathbb{M}} q_m(x_i, \mathbf{x}) = 1$ with $q_{m_i}(x_i, \mathbf{x}) = 0$. An example is provided below.

**Example 3 (Mutation kernel given by a transition matrix).** A natural example is when the particle to modify is uniformly chosen among
all particles, and then the change of label is done according to a transition matrix with coefficients $(p_{m,m'})_{m,m' \in \mathbb{M}}$. This corresponds to the choices

$$s(x_i, \mathbf{x}) = \frac{1}{n(\mathbf{x})} \quad \text{and} \quad q_m(x_i, \mathbf{x}) = p_{m_i, m},$$

where $p_{m,m} = 0$ and for all $m \in \mathbb{M}$, $\sum_{m' \in \mathbb{M}} p_{m,m'} = 1$.

Finally, for the death transition kernel, we set for any $\mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\}$ and any $F \in \mathcal{E}$, $\nu_\delta(\mathbf{x}, F) = \sum_{i=1}^{n(\mathbf{x})} \mathbf{1}_{\mathbf{x} \setminus x_i \in F}$, so that this transition only concerns the death of an existing particle. For the density,

$$k_\delta(\mathbf{x}, \mathbf{y}) = \begin{cases} \omega(x_i, x) & \text{if } \mathbf{y} = \mathbf{x} \setminus x_i, \\ 0 & \text{otherwise,} \end{cases}$$

where $\omega(x_i, \mathbf{x}) \in [0, 1]$ is the probability that the particle $x_i$ in $\mathbf{x}$ dies, with $\sum_{i=1}^{n(\mathbf{x})} \omega(x_i, \mathbf{x}) = 1$. A simple example is the uniform death where $\omega(x_i, \mathbf{x}) = 1/n(\mathbf{x})$ for any $i$.

### 2.3.3 The inter-jumps motion

Remember that between two jumps, the BDMM process $(\mathbf{X}_t)_{t \ge 0}$ has a constant cardinality $n = n(\mathbf{X}_t)$ and constant labels in $\mathbb{M}$. To specify the inter-jumps motion of $(\mathbf{X}_t)_{t \ge 0}$, it is then enough to define the dynamics of a process $(\mathbf{Y}_t^n)_{t \ge 0}$ on each subspace $E_n$ where

$$\mathbf{Y}_t^n = \{(z_{1,t}, m_1), \dots, (z_{n,t}, m_n)\}.$$

Accordingly, the position and the continuous mark of each particle may move, that is $z_{i,t} \in \Lambda$ may move, but the label $m_i$ will remain constant. To define such dynamics, we first need to come back to a standard system of $n$ ordered (labelled) particles in $(\Lambda \times \mathbb{M})^n$, that is

$$\tilde{\mathbf{Y}}_t^n = ((z_{1,t}, m_1), \dots, (z_{n,t}, m_n)).$$

The full definition of the move process $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ then follows the three steps:

1. Define the dynamics of $\tilde{\mathbf{Y}}_t^n$ on $(\Lambda \times \mathbb{M})^n$ thanks to a system of stochastic differential equations, as $(M^{|n})$ specified below, where the motion acts only in $\Lambda^n$.
all particles, and then the change of label is done according to a transition matrix with coefficients $(p_{m,m'})_{m,m' \in \mathbb{M}}$. This corresponds to the choices

$$s(x_i, \mathbf{x}) = \frac{1}{n(\mathbf{x})} \quad \text{and} \quad q_m(x_i, \mathbf{x}) = p_{m_i, m},$$

where $p_{m,m} = 0$ and for all $m \in \mathbb{M}$, $\sum_{m' \in \mathbb{M}} p_{m,m'} = 1$.

Finally, for the death transition kernel, we set for any $\mathbf{x} = \{x_1, \dots, x_{n(\mathbf{x})}\}$ and any $F \in \mathcal{E}$, $\nu_\delta(\mathbf{x}, F) = \sum_{i=1}^{n(\mathbf{x})} \mathbf{1}_{\mathbf{x}\setminus x_i \in F}$, so that this transition only concerns the death of an existing particle. For the density,

$$k_\delta(\mathbf{x}, \mathbf{y}) = \begin{cases} \omega(x_i, x) & \text{if } \mathbf{y} = \mathbf{x} \setminus x_i, \\ 0 & \text{otherwise,} \end{cases}$$

where $\omega(x_i, \mathbf{x}) \in [0, 1]$ is the probability that the particle $x_i$ in $\mathbf{x}$ dies, with $\sum_{i=1}^{n(\mathbf{x})} \omega(x_i, \mathbf{x}) = 1$. A simple example is the uniform death where $\omega(x_i, \mathbf{x}) = 1/n(\mathbf{x})$ for any $i$.

### 2.3.3 The inter-jumps motion

Remember that between two jumps, the BDMM process $(\mathbf{X}_t)_{t \ge 0}$ has a constant cardinality $n = n(\mathbf{X}_t)$ and constant labels in $\mathbb{M}$. To specify the inter-jumps motion of $(\mathbf{X}_t)_{t \ge 0}$, it is then enough to define the dynamics of a process $(\mathbf{Y}_t^n)_{t \ge 0}$ on each subspace $E_n$ where

$$\mathbf{Y}_t^n = \{(z_{1,t}, m_1), \dots, (z_{n,t}, m_n)\}.$$

Accordingly, the position and the continuous mark of each particle may move, that is $z_{i,t} \in \Lambda$ may move, but the label $m_i$ will remain constant. To define such dynamics, we first need to come back to a standard system of $n$ ordered (labelled) particles in $(\Lambda \times \mathbb{M})^n$, that is

$$\tilde{\mathbf{Y}}_t^n = ((z_{1,t}, m_1), \dots, (z_{n,t}, m_n)).$$

The full definition of the move process $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ then follows the three steps:

1. Define the dynamics of $\tilde{\mathbf{Y}}_t^n$ on $(\Lambda \times \mathbb{M})^n$ thanks to a system of stochastic differential equations, as $(M^{|n})$ specified below, where the motion acts only in $\Lambda^n$.
2. Provided this system yields a solution whose distribution satisfies the permutation equivariance property (see below), deduce $\mathbf{Y}_t^{|n|} = \Pi_n(\tilde{\mathbf{Y}}_t^{|n|})$ on $E_n$, where $\Pi_n$ is given by (1).

3. Then define $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ by $\mathbf{Y}_t = \sum_{n \ge 0} \mathbf{Y}_t^{|n|} \mathbf{1}_{\{\mathbf{Y}_0 \in E_n\}}$.

This construction and the following facts are detailed in [14]. The equivariance property means that for any permutation $\pi$ of $\{1, \dots, n\}$, the law of $\tilde{\mathbf{Y}}_t^{|n|} = ((z_{1,t}, m_1), \dots, (z_{n,t}, m_n))$ given $\tilde{\mathbf{Y}}_0^{|n|} = ((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)}))$ is the same as the distribution of $((z_{\pi(1),t}, m_{\pi(1)}), \dots, (z_{\pi(n),t}, m_{\pi(n)}))$ given $\tilde{\mathbf{Y}}_0^{|n|} = ((z_1, m_1), \dots, (z_n, m_n))$. This property says that if we rearrange the ordering of the coordinates of the initial state, the ordering of the solution is rearranged in the same way, so that it makes sense to deduce $\mathbf{Y}_t^{|n|}$ from $\tilde{\mathbf{Y}}_t^{|n|}$. Under this assumption, the Markov process $\mathbf{Y}$ is well-defined on $E$ and its transition kernel $Q_t^{\mathbf{Y}}$, defined for any bounded and measurable function $f$ on $E$ by $Q_t^{\mathbf{Y}} f(x) = \mathbb{E}[f(\mathbf{Y}_t)|\mathbf{Y}_0 = x]$, reads

$$Q_t^{\mathbf{Y}} f(x) = \sum_{n \ge 0} Q_t^{\tilde{\mathbf{Y}}^{|n|}} (f \circ \Pi_n)((x_1, \dots, x_n)) \mathbf{1}_{x \in E_n},$$

where $Q_t^{\tilde{\mathbf{Y}}^{|n|}}$ denotes the transition kernel of $\tilde{\mathbf{Y}}^{|n|}$ in $(\Lambda \times \mathbb{M})^n$. Moreover, $\mathbf{Y}$ is continuous in $(E, d_1)$ whenever for any $n$, $\tilde{\mathbf{Y}}^{|n|}$ is continuous in $(\Lambda \times \mathbb{M})^n$ for the usual Euclidean distance.

As a consequence of this construction, the definition of $(\mathbf{Y}_t)_{t \ge 0}$ boils down to the first step above, that is the specification of the system of SDE $(M^{|n|})$, provided the latter is permutation equivariant. We assume in this paper that $\tilde{\mathbf{Y}}_t^{|n|} = (\mathbf{Z}_t, \mathbf{m})$, starting at $t=0$ from $(\mathbf{z}, \mathbf{m})$ where $\mathbf{z} = (z_1, \dots, z_n)$, is the solution of

$$M^{|n|}(\mathbf{z}, \mathbf{m}) : \begin{cases} dz_{i,t} = v_{i,n}(\mathbf{Z}_t, \mathbf{m})dt + \sigma_{i,n}(\mathbf{Z}_t, \mathbf{m})dB_{i,t}, & t \ge 0, \quad i = 1, \dots, n, \\ z_{i,0} = z_i, & i = 1, \dots, n. \end{cases}$$

Here the drift functions $v_{1,n}, \dots, v_{n,n}$ take their values in $\mathbb{R}^d$, the diffusion coefficients $\sigma_{1,n}, \dots, \sigma_{n,n}$ are $(d, d)$ invertible matrices and $(B_{1,t})_{t \ge 0}, \dots, (B_{n,t})_{t \ge 0}$ are independent standard Brownian motions on $\mathbb{R}^d$. We assume that the functions $v_{i,n}$ and $\sigma_{i,n}$ are globally Lipschitzian, so that a strong solution exists. If $\Lambda \ne \mathbb{R}^d$, we further assume that edge conditions (reflective or periodic) are added to ensure that the solution stays in $(\Lambda \times \mathbb{M})^n$, see for instance [7].
2. Provided this system yields a solution whose distribution satisfies the permutation equivariance property (see below), deduce $\mathbf{Y}_t^{|n|} = \Pi_n(\tilde{\mathbf{Y}}_t^{|n|})$ on $E_n$, where $\Pi_n$ is given by (1).

3. Then define $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ by $\mathbf{Y}_t = \sum_{n \ge 0} \mathbf{Y}_t^{|n|} \mathbf{1}_{\{\mathbf{Y}_0 \in E_n\}}$.

This construction and the following facts are detailed in [14]. The equivariance property means that for any permutation $\pi$ of $\{1, \dots, n\}$, the law of $\tilde{\mathbf{Y}}_t^{|n|} = ((z_{1,t}, m_1), \dots, (z_{n,t}, m_n))$ given $\tilde{\mathbf{Y}}_0^{|n|} = ((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)}))$ is the same as the distribution of $((z_{\pi(1),t}, m_{\pi(1)}), \dots, (z_{\pi(n),t}, m_{\pi(n)}))$ given $\tilde{\mathbf{Y}}_0^{|n|} = ((z_1, m_1), \dots, (z_n, m_n))$. This property says that if we rearrange the ordering of the coordinates of the initial state, the ordering of the solution is rearranged in the same way, so that it makes sense to deduce $\mathbf{Y}_t^{|n|}$ from $\tilde{\mathbf{Y}}_t^{|n|}$. Under this assumption, the Markov process $\mathbf{Y}$ is well-defined on $E$ and its transition kernel $Q_t^{\mathbf{Y}}$, defined for any bounded and measurable function $f$ on $E$ by $Q_t^{\mathbf{Y}} f(x) = \mathbb{E}[f(\mathbf{Y}_t)|\mathbf{Y}_0 = x]$, reads

$$Q_t^{\mathbf{Y}} f(x) = \sum_{n \ge 0} Q_t^{\tilde{\mathbf{Y}}^{|n|}} (f \circ \Pi_n) ((x_1, \dots, x_n)) \mathbf{1}_{x \in E_n},$$

where $Q_t^{\tilde{\mathbf{Y}}^{|n|}}$ denotes the transition kernel of $\tilde{\mathbf{Y}}^{|n|}$ in $(\Lambda \times \mathbb{M})^n$. Moreover, $\mathbf{Y}$ is continuous in $(E, d_1)$ whenever for any $n$, $\tilde{\mathbf{Y}}^{|n|}$ is continuous in $(\Lambda \times \mathbb{M})^n$ for the usual Euclidean distance.

As a consequence of this construction, the definition of $(\mathbf{Y}_t)_{t \ge 0}$ boils down to the first step above, that is the specification of the system of SDE $(M^{|n|})$, provided the latter is permutation equivariant. We assume in this paper that $\tilde{\mathbf{Y}}_t^{|n|} = (\mathbf{Z}_t, \mathbf{m})$, starting at $t = 0$ from $(\mathbf{z}, \mathbf{m})$ where $\mathbf{z} = (z_1, \dots, z_n)$, is the solution of

$$M^{|n|}(\mathbf{z}, \mathbf{m}) : \begin{cases} dz_{i,t} = v_{i,n}(\mathbf{Z}_t, \mathbf{m})dt + \sigma_{i,n}(\mathbf{Z}_t, \mathbf{m})dB_{i,t}, & t \ge 0, \quad i = 1, \dots, n, \\ z_{i,0} = z_i, & i = 1, \dots, n. \end{cases}$$

Here the drift functions $v_{1,n}, \dots, v_{n,n}$ take their values in $\mathbb{R}^d$, the diffusion coefficients $\sigma_{1,n}, \dots, \sigma_{n,n}$ are $(d, d)$ invertible matrices and $(B_{1,t})_{t \ge 0}, \dots, (B_{n,t})_{t \ge 0}$ are independent standard Brownian motions on $\mathbb{R}^d$. We assume that the functions $v_{i,n}$ and $\sigma_{i,n}$ are globally Lipschitzian, so that a strong solution exists. If $\Lambda \neq \mathbb{R}^d$, we further assume that edge conditions (reflective or periodic) are added to ensure that the solution stays in $(\Lambda \times \mathbb{M})^n$, see for instance [7].
The permutation equivariance property is ensured if for any $i = 1, \dots, n$ and any $\pi \in S_n$,

$$ \begin{cases} v_{i,n}((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)})) = v_{\pi(i),n}((z_1, m_1), \dots, (z_n, m_n)), \\ \sigma_{i,n}((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)})) = \sigma_{\pi(i),n}((z_1, m_1), \dots, (z_n, m_n)). \end{cases} $$

This is for instance the case if $v_{i,n}(\mathbf{Z}, \mathbf{m}) = v(z_i, m_i)$ and $\sigma_{i,n}(\mathbf{Z}, \mathbf{m}) = \sigma(z_i, m_i)$ for some functions $v$ and $\sigma$, a situation where there is no interaction between the particles, as considered in the microbiological applications in [6] and [2]. An example that includes interactions through a potential function is provided in the following example.

**Example 4 (Langevin diffusion).** In this example the motion of each particle depends on an interaction force with the other particles, driven by a pair potential functions $\Phi_{m,m'}$, $m, m' \in \mathbb{M}$, as in Example 2. Specifically, in this model $\sigma_i(\mathbf{Z}, \mathbf{m}) = \sigma_{m_i}$ and

$$ v_{i,n}(\mathbf{Z}, \mathbf{m}) = -\sum_{j \neq i} \Phi_{m_i, m_j}(z_i - z_j). $$

For existence, each pair potential $\Phi_{m,m'}$ must be smooth enough. We refer to [7] and [14] for details and examples.

For later purposes, let us specify the likelihood of $\mathbf{Y}_t^{[n]}$ constructed as above. Let us first rewrite $(M^{[n]})$ so that it takes the form of a standard stochastic differential equation. Denote by $\bar{\sigma}_n$ the block diagonal matrix of size $(nd, nd)$ formed by the $\sigma_{i,n}$ matrices, and by $\bar{v}_n$ the vector of size $nd$ formed by the concatenation of the $v_{i,n}$ vectors. Then, by denoting $B_t$ the vector formed by the concatenation of the vectors $B_{i,t}$, we have

$$ M^{[n]}(\mathbf{z}, \mathbf{m}) : \begin{cases} d\mathbf{Z}_t = \bar{v}_n(\mathbf{Z}_t, \mathbf{m})dt + \bar{\sigma}_n(\mathbf{Z}_t, \mathbf{m})dB_t, & t \ge 0, \\ \mathbf{Z}_0 = \mathbf{z}. \end{cases} \quad (4) $$

It is well known, cf [16, 13], that the Radon-Nikodym density of the solution $(\mathbf{Z}_t)_{t \ge 0}$ of $M^{[n]}(\mathbf{z}, \mathbf{m})$ with respect to the reference process $\mathbf{U}_t = \mathbf{z} + \int_0^t \bar{\sigma}_n(\mathbf{U}_s, \mathbf{m})dB_s$, on the interval $[0, t]$ reads

$$ L(\mathbf{Z}_{[0,t]}, \mathbf{m}) = \exp \left( \int_0^t \bar{v}_n(\mathbf{Z}_s, \mathbf{m})^T a_n^{-1}(\mathbf{Z}_s, \mathbf{m}) d\mathbf{Z}_s - \frac{1}{2} \int_0^t \bar{v}_n(\mathbf{Z}_s, \mathbf{m})^T a_n^{-1}(\mathbf{Z}_s, \mathbf{m}) \bar{v}_n(\mathbf{Z}_s, \mathbf{m}) ds \right), \quad (5) $$
The permutation equivariance property is ensured if for any $i = 1, \dots, n$ and any $\pi \in S_n$,

$$ \left\{ \begin{array}{l} v_{i,n}((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)})) = v_{\pi(i),n}((z_1, m_1), \dots, (z_n, m_n)), \\ \sigma_{i,n}((z_{\pi(1)}, m_{\pi(1)}), \dots, (z_{\pi(n)}, m_{\pi(n)})) = \sigma_{\pi(i),n}((z_1, m_1), \dots, (z_n, m_n)). \end{array} \right. $$

This is for instance the case if $v_{i,n}(\mathbf{Z}, \mathbf{m}) = v(z_i, m_i)$ and $\sigma_{i,n}(\mathbf{Z}, \mathbf{m}) = \sigma(z_i, m_i)$ for some functions $v$ and $\sigma$, a situation where there is no interaction between the particles, as considered in the microbiological applications in [6] and [2]. An example that includes interactions through a potential function is provided in the following example.

**Example 4 (Langevin diffusion).** In this example the motion of each particle depends on an interaction force with the other particles, driven by a pair potential functions $\Phi_{m,m'}$, $m, m' \in \mathbb{M}$, as in Example 2. Specifically, in this model $\sigma_i(\mathbf{Z}, \mathbf{m}) = \sigma_{m_i}$ and

$$ v_{i,n}(\mathbf{Z}, \mathbf{m}) = -\sum_{j \neq i} \Phi_{m_i, m_j}(z_i - z_j). $$

For existence, each pair potential $\Phi_{m,m'}$ must be smooth enough. We refer to [7] and [14] for details and examples.

For later purposes, let us specify the likelihood of $\mathbf{Y}_t^{[n]}$ constructed as above. Let us first rewrite $(M^{[n]})$ so that it takes the form of a standard stochastic differential equation. Denote by $\bar{\sigma}_n$ the block diagonal matrix of size $(nd, nd)$ formed by the $\sigma_{i,n}$ matrices, and by $\bar{v}_n$ the vector of size $nd$ formed by the concatenation of the $v_{i,n}$ vectors. Then, by denoting $B_t$ the vector formed by the concatenation of the vectors $B_{i,t}$, we have

$$ M^{[n]}(\mathbf{z}, \mathbf{m}) : \left\{ \begin{array}{l} d\mathbf{Z}_t = \bar{v}_n(\mathbf{Z}_t, \mathbf{m})dt + \bar{\sigma}_n(\mathbf{Z}_t, \mathbf{m})dB_t, \quad t \ge 0, \\ \mathbf{Z}_0 = \mathbf{z}. \end{array} \right. \quad (4) $$

It is well known, cf [16, 13], that the Radon-Nikodym density of the solution $(\mathbf{Z}_t)_{t \ge 0}$ of $M^{[n]}(\mathbf{z}, \mathbf{m})$ with respect to the reference process $\mathbf{U}_t = \mathbf{z} + \int_0^t \bar{\sigma}_n(\mathbf{U}_s, \mathbf{m})dB_s$, on the interval $[0, t]$ reads

$$ L(\mathbf{Z}_{[0,t]}, \mathbf{m}) = \exp \left( \int_0^t \bar{v}_n(\mathbf{Z}_s, \mathbf{m})^T a_n^{-1}(\mathbf{Z}_s, \mathbf{m}) d\mathbf{Z}_s - \frac{1}{2} \int_0^t \bar{v}_n(\mathbf{Z}_s, \mathbf{m})^T a_n^{-1}(\mathbf{Z}_s, \mathbf{m}) \bar{v}_n(\mathbf{Z}_s, \mathbf{m}) ds \right), \quad (5) $$
where $a_n(\mathbf{z}, \mathbf{m}) = \bar{\sigma}_n(\mathbf{z}, \mathbf{m})\bar{\sigma}_n^T(\mathbf{z}, \mathbf{m})$. This gives us the density of the process $\tilde{\mathbf{Y}}_t^n = (\mathbf{Z}_t, \mathbf{m})$ on $[0, t[$ with respect to the reference process $(\mathbf{U}_t, \mathbf{m})_{t\ge0}$ that belongs to $(\Lambda \times \mathbb{M})^n$. Finally, by our construction above, if $(\mathbf{Z}_t, \mathbf{m})$ is such that $\mathbf{Y}_t^n = \Pi_n((\mathbf{Z}_t, \mathbf{m}))$, then the density of $\mathbf{Y}_t^n$ with respect to $\Pi_n((\mathbf{U}_t, \mathbf{m}))$ on $[0, t]$ takes the same form and does not depend on the ordering chosen to define $\mathbf{Z}_t$, $\mathbf{z}$ and $\mathbf{m}$ from $\mathbf{Y}_t^n$, thanks to the permutation equivariance property.

# 3 LAN property

## 3.1 Likelihood of the BDMM process

As described in the previous section, the BDMM process $(\mathbf{X}_t)_{t\ge0}$ depends on the intensity functions $\beta$, $\delta$, $\tau$, assumed to be continuous on $E$, on kernel densities for the births, the deaths and the mutations, denoted by $k_\beta(\mathbf{x}, \mathbf{y})$, $k_\delta(\mathbf{x}, \mathbf{y})$ and $k_\tau(\mathbf{x}, \mathbf{y})$, as detailed in Section 2.3.2, and on a continuous Markov diffusion model on $E$ that drives the inter-jump motion, see Section 2.3.3.

Remember that between two jumps $T_i$ and $T_{i+1}$, $\mathbf{X}_t = \mathbf{Y}_{t-T_i}^{(i)}$ where $\mathbf{Y}^{(i)}$ has the same law as $\mathbf{Y}$. Write $\mathbf{Y}_t^{(i)} = \{(\mathbf{Z}_t^{(i)}, \mathbf{m}^{(i)})\}$ where $\mathbf{m}^{(i)}$ does not depend on $t$, as required for the labels of the continuous Markov process that drives the inter-jumps motion. As assumed in Section 2.3.3, $\mathbf{Z}^{(i)}$ is the solution of $M^{|n}(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})$ given by (4), where $(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})$ are such that $\mathbf{X}_{T_i} = \{(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})\}$ and $n = n(\mathbf{X}_{T_i})$. The likelihood of the inter-jump motion between $T_i$ and $T_{i+1}$, given $T_i$, $T_{i+1}$ and $\mathbf{X}_{T_i}$, is thus $L(\mathbf{Z}_{[0,T_{i+1}-T_j]}^{(i)}, \mathbf{m}^{(i)})$ given by (5). For simplicity we will simply write the latter $L(\mathbf{X}_{[T_i,T_{i+1}]}$ in the following, since $\mathbf{X}_t = \{(\mathbf{Z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ for all $t \in [T_i, T_{i+1}]$.

Assume that all features of the BDMM process depend on a parameter $\vartheta \in \Theta \subset \mathbb{R}^\ell$ for a given $\ell > 0$. To emphasize this dependence, we introduce the argument $\vartheta$ into the notation of each of them. For example we will write $\beta(\vartheta, \mathbf{x})$ instead of $\beta(\mathbf{x})$, $k_\beta(\vartheta, \mathbf{x}, \mathbf{y})$ instead of $k_\beta(\mathbf{x}, \mathbf{y})$, $L(\vartheta, \mathbf{X}_{[T_i, T_{i+1}.]})$ instead of $L(\mathbf{X}_{[T_i, T_{i+1}.]})$, and so on. Note however that in (5), $\vartheta$ only appears in the drift coefficient $\bar{v}_n$, i.e. $\bar{v}_n(\mathbf{X}_s) = \bar{v}_n(\vartheta, \mathbf{X}_s)$, but not in the diffusion coefficient $\bar{\sigma}_n$ which is part of the reference process. This is indeed a common fact that for continuous time observations, the diffusion coefficient can be considered to be known [13].
where $a_n(\mathbf{z}, \mathbf{m}) = \bar{\sigma}_n(\mathbf{z}, \mathbf{m})\bar{\sigma}_n^T(\mathbf{z}, \mathbf{m})$. This gives us the density of the process $\tilde{\mathbf{Y}}_t^n = (\mathbf{Z}_t, \mathbf{m})$ on $[0, t]$ with respect to the reference process $(\mathbf{U}_t, \mathbf{m})_{t \ge 0}$ that belongs to $(\Lambda \times \mathbb{M})^n$. Finally, by our construction above, if $(\mathbf{Z}_t, \mathbf{m})$ is such that $\mathbf{Y}_t^n = \Pi_n((\mathbf{Z}_t, \mathbf{m}))$, then the density of $\mathbf{Y}_t^n$ with respect to $\Pi_n((\mathbf{U}_t, \mathbf{m}))$ on $[0, t]$ takes the same form and does not depend on the ordering chosen to define $\mathbf{Z}_t$, $\mathbf{z}$ and $\mathbf{m}$ from $\mathbf{Y}_t^n$, thanks to the permutation equivariance property.

# 3 LAN property

## 3.1 Likelihood of the BDMM process

As described in the previous section, the BDMM process $(\mathbf{X}_t)_{t \ge 0}$ depends on the intensity functions $\beta$, $\delta$, $\tau$, assumed to be continuous on $E$, on kernel densities for the births, the deaths and the mutations, denoted by $k_\beta(\mathbf{x}, \mathbf{y})$, $k_\delta(\mathbf{x}, \mathbf{y})$ and $k_\tau(\mathbf{x}, \mathbf{y})$, as detailed in Section 2.3.2, and on a continuous Markov diffusion model on $E$ that drives the inter-jump motion, see Section 2.3.3.

Remember that between two jumps $T_i$ and $T_{i+1}$, $\mathbf{X}_t = \mathbf{Y}_{t-T_i}^{(i)}$ where $\mathbf{Y}^{(i)}$ has the same law as $\mathbf{Y}$. Write $\mathbf{Y}_t^{(i)} = \{(\mathbf{Z}_t^{(i)}, \mathbf{m}^{(i)})\}$ where $\mathbf{m}^{(i)}$ does not depend on $t$, as required for the labels of the continuous Markov process that drives the inter-jumps motion. As assumed in Section 2.3.3, $\mathbf{Z}^{(i)}$ is the solution of $M^{|n}(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})$ given by (4), where $(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})$ are such that $\mathbf{X}_{T_i} = \{(\mathbf{z}^{(i)}, \mathbf{m}^{(i)})\}$ and $n = n(\mathbf{X}_{T_i})$. The likelihood of the inter-jump motion between $T_i$ and $T_{i+1}$, given $T_i$, $T_{i+1}$ and $\mathbf{X}_{T_i}$, is thus $L(\mathbf{Z}_{[0,T_{i+1}-T_j]}^{(i)}, \mathbf{m}^{(i)})$ given by (5). For simplicity we will simply write the latter $L(\mathbf{X}_{[T_i,T_{i+1}]}$ in the following, since $\mathbf{X}_t = \{(\mathbf{Z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ for all $t \in [T_i, T_{i+1}]$.

Assume that all features of the BDMM process depend on a parameter $\vartheta \in \Theta \subset \mathbb{R}^\ell$ for a given $\ell > 0$. To emphasize this dependence, we introduce the argument $\vartheta$ into the notation of each of them. For example we will write $\beta(\vartheta, \mathbf{x})$ instead of $\beta(\mathbf{x})$, $k_\beta(\vartheta, \mathbf{x}, \mathbf{y})$ instead of $k_\beta(\mathbf{x}, \mathbf{y})$, $L(\vartheta, \mathbf{X}_{[T_i,T_{i+1}.]})$ instead of $L(\mathbf{X}_{[T_i,T_{i+1}.]})$, and so on. Note however that in (5), $\vartheta$ only appears in the drift coefficient $\bar{v}_n$, i.e. $\bar{v}_n(\mathbf{X}_s) = \bar{v}_n(\vartheta, \mathbf{X}_s)$, but not in the diffusion coefficient $\bar{\sigma}_n$ which is part of the reference process. This is indeed a common fact that for continuous time observations, the diffusion coefficient can be considered to be known [13].
Let us denote the set of birth times up to time $t$ by

$$ \mathcal{B}_t = \{i \in \mathbb{N}^*, T_i \le t \text{ and } T_i \text{ is a birth time}\}, $$

and similarly $\mathcal{D}_t$ and $\mathcal{T}_t$ for the set of death times and mutation times up to time $t$.

The following theorem provides the likelihood of the BDMM process. When there is no move and no mutation, we recover the formula given in [22] for their applications to the dynamics of dunes. It also has a consistent expression with the likelihood of a system of particles with killing and jumps, as derived in [18]. The justifications, detailed in Section A, are however different and simpler, as they only rely on successive backward applications of conditional expectations.

**Theorem 1.** Let $(\mathbf{X}_t)_{t \ge 0}$ be a BDMM parameterized by $\vartheta \in \Theta$, according to the above conditions, and observed in continuous time on $[0, t]$, for $t > 0$. Then, the likelihood $\mathcal{L}_t(\vartheta)$ is expressed as follows:

$$ \mathcal{L}_t(\vartheta) = \mathcal{L}_t^B(\vartheta) \mathcal{L}_t^D(\vartheta) \mathcal{L}_t^T(\vartheta) \mathcal{L}_t^{move}(\vartheta), $$

with

$$ \mathcal{L}_t^B(\vartheta) = \exp \left( - \int_0^t \beta(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{B}_t} k_\beta(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \beta(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^D(\vartheta) = \exp \left( - \int_0^t \delta(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{D}_t} k_\delta(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \delta(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^T(\vartheta) = \exp \left( - \int_0^t \tau(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{T}_t} k_\tau(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \tau(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^{move}(\vartheta) = L(\vartheta, \mathbf{X}_{[T_{N_t}, t]}) \prod_{i=0}^{N_t-1} L(\vartheta, \mathbf{X}_{[T_i, T_{i+1}]}). $$

Note that due to the Markov nature of the process, the likelihood is a simple product. As a consequence, inference on a specific aspect of the process can be done independently on the other characteristics, provided the parameter is different in each characteristic. As an illustration, we will infer the birth kernel $k_\beta$ in our application in Section 4, without needing to specify or estimate the other terms.
Let us denote the set of birth times up to time $t$ by

$$ \mathcal{B}_t = \{i \in \mathbb{N}^*, T_i \le t \text{ and } T_i \text{ is a birth time}\}, $$

and similarly $\mathcal{D}_t$ and $\mathcal{T}_t$ for the set of death times and mutation times up to time $t$.

The following theorem provides the likelihood of the BDMM process. When there is no move and no mutation, we recover the formula given in [22] for their applications to the dynamics of dunes. It also has a consistent expression with the likelihood of a system of particles with killing and jumps, as derived in [18]. The justifications, detailed in Section A, are however different and simpler, as they only rely on successive backward applications of conditional expectations.

**Theorem 1.** Let $(\mathbf{X}_t)_{t \ge 0}$ be a BDMM parameterized by $\vartheta \in \Theta$, according to the above conditions, and observed in continuous time on $[0, t]$, for $t > 0$. Then, the likelihood $\mathcal{L}_t(\vartheta)$ is expressed as follows:

$$ \mathcal{L}_t(\vartheta) = \mathcal{L}_t^{\mathcal{B}}(\vartheta) \mathcal{L}_t^{\mathcal{D}}(\vartheta) \mathcal{L}_t^{\mathcal{T}}(\vartheta) \mathcal{L}_t^{\text{move}}(\vartheta), $$

with

$$ \mathcal{L}_t^{\mathcal{B}}(\vartheta) = \exp \left( - \int_0^t \beta(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{B}_t} k_{\beta}(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \beta(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^{\mathcal{D}}(\vartheta) = \exp \left( - \int_0^t \delta(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{D}_t} k_{\delta}(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \delta(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^{\mathcal{T}}(\vartheta) = \exp \left( - \int_0^t \tau(\vartheta, \mathbf{X}_s) ds \right) \prod_{i \in \mathcal{T}_t} k_{\tau}(\vartheta, \mathbf{X}_{T_{i-}}, \mathbf{X}_{T_i}) \tau(\vartheta, \mathbf{X}_{T_{i-}}), $$

$$ \mathcal{L}_t^{\text{move}}(\vartheta) = L(\vartheta, \mathbf{X}_{[T_{N_t}, t]}) \prod_{i=0}^{N_t-1} L(\vartheta, \mathbf{X}_{[T_i, T_{i+1}]}). $$

Note that due to the Markov nature of the process, the likelihood is a simple product. As a consequence, inference on a specific aspect of the process can be done independently on the other characteristics, provided the parameter is different in each characteristic. As an illustration, we will infer the birth kernel $k_{\beta}$ in our application in Section 4, without needing to specify or estimate the other terms.
## 3.2 LAN

Recall that all the features of the BDMM process depend on a parameter $\vartheta \in \Theta \subset \mathbb{R}^l$. In the following, for a multivariate function $f$, we denote by $\nabla_i f$ the gradient (or Jacobian matrix if $f$ is multivariate) with respect to the $i$-th variable. We will also denote by $B(\vartheta, \rho)$ the ball centred at $\vartheta$ with radius $\rho > 0$.

We list below the assumptions we make to obtain the local asymptotic normality of our model. They are of two kinds: the first one (H) ensures that the process is non-explosive and geometric ergodic. Conditions implying this hypothesis concern the intensity functions: it basically holds if the total intensity function $\alpha$ is bounded and the death intensity function $\delta$ sufficiently compensates for the birth intensity $\beta$. Precise conditions are given in Appendix C.1, following [14]. The other assumptions are regularity hypotheses on the way each feature of the BDMM process is parametrised. Although quite technical at first sight, we show in Section 3.3 that they are satisfied in many situations, including the examples of Section 2.3.

(H) The process is non-explosive, i.e. $N_t < \infty$ for any $t \ge 0$, and there exists a measure $\mu_\infty^\vartheta$ on $E$, $r > 0$ and $c: E \to \mathbb{R}_+$ such that for any $t > 0$, any $\mathbf{x}_0 \in E$ and any measurable and bounded function $g$ on $E$,

$$ \left| \frac{1}{t} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \left( \int_0^t g(\mathbf{X}_s) ds \right) - \int_E g(\mathbf{y}) \mu_\infty^\vartheta (d\mathbf{y}) \right| \le \frac{c(\mathbf{x}_0) \|g\|_\infty}{rt}. $$

Concerning the intensity function $\gamma$ (whether $\gamma$ is $\beta$, $\delta$ or $\tau$), we assume that it is differentiable with respect to $\vartheta$ and the two following hypotheses.

A($\gamma$) For all $\vartheta \in \Theta$, $\mathbf{x} \mapsto \frac{\|\nabla_1 \gamma(\vartheta, \mathbf{x})\|^2}{\gamma(\vartheta, \mathbf{x})} \mathbf{1}_{\gamma(\vartheta, \mathbf{x})>0}$ is bounded.

B($\gamma$) For all $\vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto f_\vartheta^\gamma(\mathbf{x}, \rho(\vartheta))\gamma(\vartheta, \mathbf{x})$ is bounded, where

$$ f_{\vartheta}^{\gamma}(\mathbf{x}, \rho) = \sup_{u \in B(\vartheta, \rho)} \left( \| \nabla_1 \gamma(u, \mathbf{x}) - \nabla_1 \gamma(\vartheta, \mathbf{x}) \|^2 \right) \frac{\mathbf{1}_{\gamma(\vartheta, \mathbf{x})>0}}{(\gamma(\vartheta, \mathbf{x}))^2}. \quad (6) $$

Similarly, we assume that the transition kernel $k_\gamma$ is differentiable with respect to $\vartheta$ and the two following hypotheses.
## 3.2 LAN

Recall that all the features of the BDMM process depend on a parameter $\vartheta \in \Theta \subset \mathbb{R}^l$. In the following, for a multivariate function $f$, we denote by $\nabla_i f$ the gradient (or Jacobian matrix if $f$ is multivariate) with respect to the $i$-th variable. We will also denote by $B(\vartheta, \rho)$ the ball centred at $\vartheta$ with radius $\rho > 0$.

We list below the assumptions we make to obtain the local asymptotic normality of our model. They are of two kinds: the first one (H) ensures that the process is non-explosive and geometric ergodic. Conditions implying this hypothesis concern the intensity functions: it basically holds if the total intensity function $\alpha$ is bounded and the death intensity function $\delta$ sufficiently compensates for the birth intensity $\beta$. Precise conditions are given in Appendix C.1, following [14]. The other assumptions are regularity hypotheses on the way each feature of the BDMM process is parametrised. Although quite technical at first sight, we show in Section 3.3 that they are satisfied in many situations, including the examples of Section 2.3.

(H) The process is non-explosive, i.e. $N_t < \infty$ for any $t \ge 0$, and there exists a measure $\mu_\infty^\vartheta$ on $E$, $r > 0$ and $c: E \to \mathbb{R}_+$ such that for any $t > 0$, any $\mathbf{x}_0 \in E$ and any measurable and bounded function $g$ on $E$,

$$ \left| \frac{1}{t} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \left( \int_0^t g(\mathbf{X}_s) ds \right) - \int_E g(\mathbf{y}) \mu_\infty^\vartheta (d\mathbf{y}) \right| \le \frac{c(\mathbf{x}_0) \|g\|_\infty}{rt}. $$

Concerning the intensity function $\gamma$ (whether $\gamma$ is $\beta$, $\delta$ or $\tau$), we assume that it is differentiable with respect to $\vartheta$ and the two following hypotheses.

A($\gamma$) For all $\vartheta \in \Theta$, $\mathbf{x} \mapsto \frac{\|\nabla_1 \gamma(\vartheta, \mathbf{x})\|^2}{\gamma(\vartheta, \mathbf{x})} \mathbf{1}_{\gamma(\vartheta, \mathbf{x})>0}$ is bounded.

B($\gamma$) For all $\vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto f_\vartheta^\gamma(\mathbf{x}, \rho(\vartheta))\gamma(\vartheta, \mathbf{x})$ is bounded, where

$$ f_{\vartheta}^{\gamma}(\mathbf{x}, \rho) = \sup_{u \in B(\vartheta, \rho)} \left( \|\nabla_1 \gamma(u, \mathbf{x}) - \nabla_1 \gamma(\vartheta, \mathbf{x})\|^2 \right) \frac{\mathbf{1}_{\gamma(\vartheta, \mathbf{x})>0}}{(\gamma(\vartheta, \mathbf{x}))^2}. \quad (6) $$

Similarly, we assume that the transition kernel $k_\gamma$ is differentiable with respect to $\vartheta$ and the two following hypotheses.
A(k, γ) For all $\vartheta \in \Theta$, $\mathbf{x} \mapsto \gamma(\vartheta, \mathbf{x}) \int_E \frac{\|\nabla_1 k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})} \mathbf{1}_{k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\gamma(\mathbf{x}, d\mathbf{y})$ is bounded.

B(k, γ) For all $\vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto f_{\vartheta}^{k_\gamma}(\mathbf{x}, \rho(\vartheta))\gamma(\vartheta, \mathbf{x})$ is bounded, where

$$f_{\vartheta}^{k_{\gamma}}(\mathbf{x}, \rho) = \int_{E} \sup_{u \in B(\vartheta, \rho)} \left( \|\nabla_1 k_{\gamma}(u, \mathbf{x}, \mathbf{y}) - \nabla_1 k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})\|^2 \right) \frac{\mathbf{1}_{k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})>0}}{k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})} \nu_{\gamma}(\mathbf{x}, d\mathbf{y}). \quad (7)$$

Finally, concerning the inter-jump motion given as a solution of (4), we assume that its drift function $\bar{v}_n$ is differentiable with respect to $\vartheta$ and:

A\_move For all $n \ge 1$, $\forall \vartheta \in \Theta$, $\mathbf{x} \mapsto \|\bar{\sigma}_n(\mathbf{x})^{-1}\nabla_1 \bar{v}_n(\vartheta, \mathbf{x})\|$ is bounded.

B\_move For all $n \ge 1$, $\forall \vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto \|\bar{\sigma}_n(\mathbf{x})^{-1}\|^2 f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho(\vartheta))$ is bounded, where

$$f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho) = \sup_{u \in B(\vartheta, \rho)} \|\nabla_1 \bar{v}_n(u, \mathbf{x}) - \nabla_1 \bar{v}_n(\vartheta, \mathbf{x})\|^2. \quad (8)$$

In order to state the LAN property, we introduce, for any $\vartheta \in \Theta$ and $N \in \mathbb{N}^*$, the following $\mathcal{F}_{tN}$-martingales, along with their associated angle brackets. The fact that they are truly martingales and the expression of their brackets are justified in the proof of the following theorem deferred to Appendix B. We recall that $a_n = \bar{\sigma}_n \bar{\sigma}_n^T$.

$$M_{\gamma}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \int_{0}^{tN} \frac{\nabla_1 \gamma(\vartheta, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} dM_{s}^{\gamma}, \quad (9)$$

$$M_{k_{\gamma}}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \left( \int_{0}^{tN} \frac{\nabla_1 k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})}{k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})} dN_{s}^{\gamma} - \int_{0}^{tN} \int_{E} \nabla_1 k_{\gamma}(\vartheta, \mathbf{X}_{s}, \mathbf{y}) \nu_{\gamma}(\mathbf{X}_{s}, d\mathbf{y}) \gamma(\vartheta, \mathbf{X}_{s}) ds \right), \quad (10)$$

$$M_{L}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \sum_{i \ge 0} \int_{0}^{tN} \mathbf{1}_{[T_i; T_{i+1}[(s)(\nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_{s^{-}}))^T a_n^{-1}(\mathbf{X}_{s^{-}}) dM_{s-T_i}^{(i)}, \quad (11)$$
A(k, γ) For all $\vartheta \in \Theta$, $\mathbf{x} \mapsto \gamma(\vartheta, \mathbf{x}) \int_E \frac{\|\nabla_1 k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})} \mathbf{1}_{k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\gamma(\mathbf{x}, d\mathbf{y})$ is bounded.

B(k, γ) For all $\vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto f_{\vartheta}^{k_\gamma}(\mathbf{x}, \rho(\vartheta))\gamma(\vartheta, \mathbf{x})$ is bounded, where

$$f_{\vartheta}^{k_{\gamma}}(\mathbf{x}, \rho) = \int_{E} \sup_{u \in B(\vartheta, \rho)} \left( \|\nabla_1 k_{\gamma}(u, \mathbf{x}, \mathbf{y}) - \nabla_1 k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})\|^2 \right) \frac{\mathbf{1}_{k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})>0}}{k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})} \nu_{\gamma}(\mathbf{x}, d\mathbf{y}). \quad (7)$$

Finally, concerning the inter-jump motion given as a solution of (4), we assume that its drift function $\bar{v}_n$ is differentiable with respect to $\vartheta$ and:

A\_move For all $n \ge 1$, $\forall \vartheta \in \Theta$, $\mathbf{x} \mapsto \|\bar{\sigma}_n(\mathbf{x})^{-1}\nabla_1 \bar{v}_n(\vartheta, \mathbf{x})\|$ is bounded.

B\_move For all $n \ge 1$, $\forall \vartheta \in \Theta$, $\exists \rho(\vartheta) > 0$ such that $\mathbf{x} \mapsto \|\bar{\sigma}_n(\mathbf{x})^{-1}\|^2 f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho(\vartheta))$ is bounded, where

$$f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho) = \sup_{u \in B(\vartheta, \rho)} \|\nabla_1 \bar{v}_n(u, \mathbf{x}) - \nabla_1 \bar{v}_n(\vartheta, \mathbf{x})\|^2. \quad (8)$$

In order to state the LAN property, we introduce, for any $\vartheta \in \Theta$ and $N \in \mathbb{N}^*$, the following $\mathcal{F}_{tN}$-martingales, along with their associated angle brackets. The fact that they are truly martingales and the expression of their brackets are justified in the proof of the following theorem deferred to Appendix B. We recall that $a_n = \bar{\sigma}_n \bar{\sigma}_n^T$.

$$M_{\gamma}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \int_{0}^{tN} \frac{\nabla_1 \gamma(\vartheta, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} dM_{s}^{\gamma}, \quad (9)$$

$$M_{k_{\gamma}}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \left( \int_{0}^{tN} \frac{\nabla_1 k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})}{k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})} dN_{s}^{\gamma} - \int_{0}^{tN} \int_{E} \nabla_1 k_{\gamma}(\vartheta, \mathbf{X}_{s}, \mathbf{y}) \nu_{\gamma}(\mathbf{X}_{s}, d\mathbf{y}) \gamma(\vartheta, \mathbf{X}_{s}) ds \right), \quad (10)$$

$$M_{L}^{N, \vartheta}(t) = \frac{1}{\sqrt{N}} \sum_{i \ge 0} \int_{0}^{tN} \mathbf{1}_{[T_i; T_{i+1}[(s)(\nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_{s^{-}}))^T a_n^{-1}(\mathbf{X}_{s^{-}}) dM_{s-T_i}^{(i)}, \quad (11)$$
where $M_t^\gamma = N_t^\gamma - \int_0^t \gamma(\vartheta, \mathbf{X}_s)ds$ and $M_s^{(i)} = (\mathbf{Z}_s^{(i)} - \mathbf{Z}_0^{(i)} - \int_0^s \bar{v}_n(\vartheta, \mathbf{Z}_u^{(i)})du) \mathbf{1}_{s \ge T_i}$
if we agree that $\mathbf{X}_t = \{(\mathbf{Z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ when $t \in [T_i, T_{i+1}]$. Their associated
angle brackets are given by

$$ \langle M_{\gamma}^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \mathbf{1}_{\gamma>0} \frac{(\nabla_1 \gamma)(\nabla_1 \gamma)^T}{\gamma} (\vartheta, \mathbf{X}_s) ds, \quad (12) $$

$$ \langle M_{k_{\gamma}}^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \int_{E} \mathbf{1}_{k_{\gamma}>0} \frac{\nabla_1 k_{\gamma} (\nabla_1 k_{\gamma})^T}{k_{\gamma}} (\vartheta, \mathbf{X}_s, \mathbf{y}) \nu_{\gamma}(\mathbf{X}_s, d\mathbf{y}) \gamma(\vartheta, \mathbf{X}_s) ds, \quad (13) $$

$$ \langle M_L^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_0^{tN} (\nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_s))^T a_n^{-1}(\mathbf{X}_s) \nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_s) ds. \quad (14) $$

Finally denoting $\mathcal{S} = \{L, \beta, \delta, \tau, k_{\beta}, k_{\delta}, k_{\tau}\}$, we introduce

$$ M^{N, \vartheta}(t) = \sum_{i \in \mathcal{S}} M_i^{N, \vartheta}(t), \quad (15) $$

the angle brackets of which reads

$$ \langle M^{N, \vartheta} \rangle (t) = \sum_{i \in \mathcal{S}} \langle M_i^{N, \vartheta} \rangle (t). \quad (16) $$

**Theorem 2.** Under the same conditions as in Theorem 1, suppose that (H), **A**($\gamma$), **B**($\gamma$), **A**($k_{\gamma}$, $\gamma$), **B**($k_{\gamma}$, $\gamma$), **A**<sub>move</sub> and **B**<sub>move</sub> are satisfied (whether $\gamma$ is $\beta$, $\delta$ or $\tau$). Then for all $\vartheta \in \mathbb{R}^l$, $t > 0$, $h \in \mathbb{R}^l$ and $N \in \mathbb{N}^*$, we have the following decomposition:

$$ \log \frac{\mathcal{L}_{tN}(\vartheta + \frac{h}{\sqrt{N}})}{\mathcal{L}_{tN}(\vartheta)} = h^T M^{N, \vartheta}(t) - \frac{1}{2} h^T \langle M^{N, \vartheta} \rangle (t) h + \text{Rem}_{N, \vartheta, h}(t), \quad (17) $$

where $\text{Rem}_{N, \vartheta, h}(t) \to 0$ in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, for any $\mathbf{x}_0 \in E$, and where $M^{N, \vartheta}$ and $\langle M^{N, \vartheta} \rangle$ are given by (15) and (16). Moreover, for all $t > 0$ and $\vartheta \in \Theta$,

$$ (M^{N, \vartheta}(t), \langle M^{N, \vartheta} \rangle(t)) \xrightarrow[N \to +\infty]{} (\mathcal{N}(0, tJ(\vartheta)), tJ(\vartheta)), \quad (18) $$

where the joint convergence takes place in distribution and where the matrix
where $M_t^\gamma = N_t^\gamma - \int_0^t \gamma(\vartheta, \mathbf{X}_s)ds$ and $M_s^{(i)} = (\mathbf{Z}_s^{(i)} - \mathbf{Z}_0^{(i)} - \int_0^s \bar{v}_n(\vartheta, \mathbf{Z}_u^{(i)})du) \mathbf{1}_{s \ge T_i}$
if we agree that $\mathbf{X}_t = \{(\mathbf{Z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ when $t \in [T_i, T_{i+1}[$. Their associated
angle brackets are given by

$$ \langle M_{\gamma}^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \mathbf{1}_{\gamma>0} \frac{(\nabla_1 \gamma)(\nabla_1 \gamma)^T}{\gamma} (\vartheta, \mathbf{X}_s) ds, \quad (12) $$

$$ \langle M_{k_{\gamma}}^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \int_{E} \mathbf{1}_{k_{\gamma}>0} \frac{\nabla_1 k_{\gamma} (\nabla_1 k_{\gamma})^T}{k_{\gamma}} (\vartheta, \mathbf{X}_s, \mathbf{y}) \nu_{\gamma}(\mathbf{X}_s, d\mathbf{y}) \gamma(\vartheta, \mathbf{X}_s) ds, \quad (13) $$

$$ \langle M_L^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_0^{tN} (\nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_s))^T a_n^{-1}(\mathbf{X}_s) \nabla_1 \bar{v}_n(\vartheta, \mathbf{X}_s) ds. \quad (14) $$

Finally denoting $\mathcal{S} = \{L, \beta, \delta, \tau, k_\beta, k_\delta, k_\tau\}$, we introduce

$$ M^{N, \vartheta}(t) = \sum_{i \in \mathcal{S}} M_i^{N, \vartheta}(t), \quad (15) $$

the angle brackets of which reads

$$ \langle M^{N, \vartheta} \rangle (t) = \sum_{i \in \mathcal{S}} \langle M_i^{N, \vartheta} \rangle (t). \quad (16) $$

**Theorem 2.** Under the same conditions as in Theorem 1, suppose that (H), **A**($\gamma$), **B**($\gamma$), **A**($k_\gamma$, $\gamma$), **B**($k_\gamma$, $\gamma$), **A**<sub>move</sub> and **B**<sub>move</sub> are satisfied (whether $\gamma$ is $\beta$, $\delta$ or $\tau$). Then for all $\vartheta \in \mathbb{R}^\ell$, $t > 0$, $h \in \mathbb{R}^\ell$ and $N \in \mathbb{N}^*$, we have the following decomposition:

$$ \log \frac{\mathcal{L}_{tN}(\vartheta + \frac{h}{\sqrt{N}})}{\mathcal{L}_{tN}(\vartheta)} = h^T M^{N, \vartheta}(t) - \frac{1}{2} h^T \langle M^{N, \vartheta} \rangle(t)h + \mathrm{Rem}_{N, \vartheta, h}(t), \quad (17) $$

where $\mathrm{Rem}_{N, \vartheta, h}(t) \to 0$ in $\mathbb{P}_{x_0}^\vartheta$-probability when $N \to +\infty$, for any $\mathbf{x}_0 \in E$, and where $M^{N, \vartheta}$ and $\langle M^{N, \vartheta} \rangle$ are given by (15) and (16). Moreover, for all $t > 0$ and $\vartheta \in \Theta$,

$$ (M^{N, \vartheta}(t), \langle M^{N, \vartheta} \rangle(t)) \xrightarrow[N \to +\infty]{} (\mathcal{N}(0, tJ(\vartheta)), tJ(\vartheta)), \quad (18) $$

where the joint convergence takes place in distribution and where the matrix
$J(\vartheta)$, of size $(\ell, \ell)$, has for expression:

$$
\begin{align}
J(\vartheta) &= \lim_{N \to +\infty} \frac{1}{t} \langle M^{N,\vartheta} \rangle(t) \tag{19} \\
&= \int_E (\nabla_1 \bar{v}_n(\vartheta, \mathbf{x}))^T a_n^{-1}(\mathbf{x}) \nabla_1 \bar{v}_n(\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}) \nonumber \\
&\quad + \sum_{\gamma \in \{\beta, \delta, \tau\}} \int_E \mathbf{1}_{\gamma>0} \frac{(\nabla_1 \gamma)(\nabla_1 \gamma)^T}{\gamma} (\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}) \nonumber \\
&\quad + \sum_{\gamma \in \{\beta, \delta, \tau\}} \int_E \int_E \mathbf{1}_{k_\gamma>0} \frac{\nabla_1 k_\gamma (\nabla_1 k_\gamma)^T}{k_\gamma} (\vartheta, \mathbf{x}, \mathbf{y}) \nu_\gamma(\mathbf{x}, d\mathbf{y}) \gamma(\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}). \nonumber
\end{align}
$$

Under additional regularity assumptions on the model, as detailed for in-
stance in [9], the LAN property implies that the maximum likelihood estima-
tor is asymptotically efficient and asymptotically Gaussian with covariance
$J(\vartheta)^{-1}$. The following corollary summarise this consequence, the proof of
which may be found in [9].

**Corollary 1.** In addition to the setting of Theorem 2, assume that the regularity conditions N2-N4 of [9, Chapter III] are satisfied. Then the maximum likelihood estimator $\hat{\vartheta}_N$, defined by $\mathcal{L}_N(\hat{\vartheta}_N) = \sup_\vartheta \mathcal{L}_N(\vartheta)$, where $\mathcal{L}_N$ is given in Theorem 1, is asymptotically efficient and satisfies

$$
\sqrt{N}(\hat{\vartheta}_N - \vartheta) \xrightarrow[N \to +\infty]{} \mathcal{N}(0, J(\vartheta)^{-1}),
$$

where the convergence is in distribution and $J(\vartheta)$ is given by (19).

Note that in Theorem 2, the length of the observed trajectory is Nt, and
the asymptotic regime stands for N large and any t > 0. We set in the above
corollary t = 1 for simplicity and considered a trajectory of size N.

In practice, the asymptotic covariance matrix $J(\vartheta)^{-1}$ is estimated by $J_N(\hat{\vartheta}_N)^{-1}$, where $\hat{\vartheta}_N$ is the maximum likelihood estimator and $J_N$ is the empirical version of $J$ given by (19), viz.

$$
J_N(\vartheta) = \langle M^{N,\vartheta} \rangle (1),
$$

where $\langle M^{N,\vartheta} \rangle$ is given by (16) and involves the terms (12), (13) and (14).
All these quantities are computable from a single trajectory over $[0, N]$, as
implemented in Section 4.
$J(\vartheta)$, of size $(\ell, \ell)$, has for expression:

$$
\begin{align}
J(\vartheta) &= \lim_{N \to +\infty} \frac{1}{t} \langle M^{N,\vartheta} \rangle(t) \tag{19} \\
&= \int_E (\nabla_1 \bar{v}_n(\vartheta, \mathbf{x}))^T a_n^{-1}(\mathbf{x}) \nabla_1 \bar{v}_n(\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}) \nonumber \\
&\quad + \sum_{\gamma \in \{\beta, \delta, \tau\}} \int_E \mathbf{1}_{\gamma>0} \frac{(\nabla_1 \gamma)(\nabla_1 \gamma)^T}{\gamma} (\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}) \nonumber \\
&\quad + \sum_{\gamma \in \{\beta, \delta, \tau\}} \int_E \int_E \mathbf{1}_{k_\gamma>0} \frac{\nabla_1 k_\gamma (\nabla_1 k_\gamma)^T}{k_\gamma} (\vartheta, \mathbf{x}, \mathbf{y}) \nu_\gamma(\mathbf{x}, d\mathbf{y}) \gamma(\vartheta, \mathbf{x}) \mu_\infty^\vartheta(d\mathbf{x}). \nonumber
\end{align}
$$

Under additional regularity assumptions on the model, as detailed for in-
stance in [9], the LAN property implies that the maximum likelihood estima-
tor is asymptotically efficient and asymptotically Gaussian with covariance
$J(\vartheta)^{-1}$. The following corollary summarise this consequence, the proof of
which may be found in [9].

**Corollary 1.** In addition to the setting of Theorem 2, assume that the regularity conditions N2-N4 of [9, Chapter III] are satisfied. Then the maximum likelihood estimator $\hat{\vartheta}_N$, defined by $\mathcal{L}_N(\hat{\vartheta}_N) = \sup_\vartheta \mathcal{L}_N(\vartheta)$, where $\mathcal{L}_N$ is given in Theorem 1, is asymptotically efficient and satisfies

$$
\sqrt{N}(\hat{\vartheta}_N - \vartheta) \xrightarrow[N \to +\infty]{} \mathcal{N}(0, J(\vartheta)^{-1}),
$$

where the convergence is in distribution and $J(\vartheta)$ is given by (19).

Note that in Theorem 2, the length of the observed trajectory is Nt, and
the asymptotic regime stands for N large and any t > 0. We set in the above
corollary t = 1 for simplicity and considered a trajectory of size N.

In practice, the asymptotic covariance matrix $J(\vartheta)^{-1}$ is estimated by $J_N(\hat{\vartheta}_N)^{-1}$, where $\hat{\vartheta}_N$ is the maximum likelihood estimator and $J_N$ is the empirical version of $J$ given by (19), viz.

$$
J_N(\vartheta) = \langle M^{N,\vartheta} \rangle (1),
$$

where $\langle M^{N,\vartheta} \rangle$ is given by (16) and involves the terms (12), (13) and (14).
All these quantities are computable from a single trajectory over $[0, N]$, as
implemented in Section 4.
## 3.3 Examples

We show in this section that the regularity conditions **A**($\gamma$), **B**($\gamma$), **A**($k_\gamma$, $\gamma$), **B**($k_\gamma$, $\gamma$), **A**<sub>move</sub> and **B**<sub>move</sub> are satisfied under mild assumptions for the examples considered in Section 2.3.

For the intensity functions, whether $\gamma(\mathbf{x}) = \gamma$ is constant or $\gamma(\mathbf{x}) = \gamma n(\mathbf{x})$, for $\gamma > 0$, and the parameter of interest is $\vartheta = \gamma$, we have $f_\gamma^\gamma = 0$ in (6), so that **B**($\gamma$) is obviously satisfied. For **A**($\gamma$), it clearly holds in the first case, while it is true in the second case if we assume that there is a maximal number of particles, i.e. $\sup_\mathbf{x} n(\mathbf{x}) < n^*$ for some $n^* > 0$. Note that the latter restriction is purely theoretical, but it is not a limitation in practice. It is also an assumption made in [15] and it amounts to the specific setting of condition (46) presented in appendix, that implies geometric ergodicity of the process, i.e. (H).

Concerning the transition kernels, the parameters of interest are typically involved in the birth kernel and/or the mutation kernel. For the birth kernel, remember that it reads as in (3), i.e. $k_\beta(\mathbf{x}, \mathbf{y}) = p_m k_\beta^m(\mathbf{x}, \mathbf{z})$ and let us examine Examples 1 and 2 for $k_\beta^m$.

**Example 1 (continued).** In this example $k_\beta^m$ is a mixture of Gaussian distributions with deviation $\sigma > 0$, so that the parameters of interest for the birth kernel are in this case $p_m, m \in \mathbb{M}$, and $\sigma$. It is not difficult to check that if $\Lambda$ is a bounded set, then all partial derivatives of $k_\beta(\mathbf{x}, \mathbf{y})$ with respect to each parameter are uniformly bounded in $\mathbf{x}$ and $\mathbf{y}$. On the other hand $k_\beta(\mathbf{x}, \mathbf{y})$ itself is lower bounded in $\mathbf{x}$ and $\mathbf{y}$ under the same setting. Consequently, **A**($k_\gamma$, $\gamma$) holds true whenever $\Lambda$ is a bounded set and the birth intensity $\beta(\mathbf{x})$ is bounded in $\mathbf{x}$. The latter is in particular true if $\beta$ satisfies the setting discussed in the previous paragraph. For **B**($k_\gamma$, $\gamma$), note that a rough control consists in upper-bounding the squared norm of the difference in (7) by twice the sum of each squared norm. Then the same bounds as previously can be used to check **B**($k_\gamma$, $\gamma$) in the same setting.

**Example 2 (continued).** Assume that $k_\beta^m(\mathbf{x}, \mathbf{z}) = e^{-\vartheta'_m S_m(\mathbf{x}, \mathbf{z})}/a^m(\mathbf{x})$, where $\vartheta_m = (\vartheta_{m,m'})_{m' \in \mathbb{M}}$, $a^m(\mathbf{x}) = \int_\Lambda e^{-\vartheta'_m S_m(\mathbf{x}, \mathbf{z})} dz$ and $S_m$ is the vector

$$S_m(\mathbf{x}, \mathbf{z}) = \left( \sum_{\substack{x_i=(z_i, m_i) \in \mathbf{x} \\ m_i=m'}} \Phi_{m,m'}(z-z_i) \right)_{m' \in \mathbb{M}}.$$
## 3.3 Examples

We show in this section that the regularity conditions **A**($\gamma$), **B**($\gamma$), **A**($k_\gamma$, $\gamma$), **B**($k_\gamma$, $\gamma$), **A**<sub>move</sub> and **B**<sub>move</sub> are satisfied under mild assumptions for the examples considered in Section 2.3.

For the intensity functions, whether $\gamma(\mathbf{x}) = \gamma$ is constant or $\gamma(\mathbf{x}) = \gamma n(\mathbf{x})$, for $\gamma > 0$, and the parameter of interest is $\vartheta = \gamma$, we have $f_\gamma^\gamma = 0$ in (6), so that **B**($\gamma$) is obviously satisfied. For **A**($\gamma$), it clearly holds in the first case, while it is true in the second case if we assume that there is a maximal number of particles, i.e. $\sup_\mathbf{x} n(\mathbf{x}) < n^*$ for some $n^* > 0$. Note that the latter restriction is purely theoretical, but it is not a limitation in practice. It is also an assumption made in [15] and it amounts to the specific setting of condition (46) presented in appendix, that implies geometric ergodicity of the process, i.e. (**H**).

Concerning the transition kernels, the parameters of interest are typically involved in the birth kernel and/or the mutation kernel. For the birth kernel, remember that it reads as in (3), i.e. $k_\beta(\mathbf{x}, \mathbf{y}) = p_m k_\beta^m(\mathbf{x}, \mathbf{z})$ and let us examine Examples 1 and 2 for $k_\beta^m$.

**Example 1 (continued).** In this example $k_\beta^m$ is a mixture of Gaussian distributions with deviation $\sigma > 0$, so that the parameters of interest for the birth kernel are in this case $p_m, m \in \mathbb{M}$, and $\sigma$. It is not difficult to check that if $\Lambda$ is a bounded set, then all partial derivatives of $k_\beta(\mathbf{x}, \mathbf{y})$ with respect to each parameter are uniformly bounded in $\mathbf{x}$ and $\mathbf{y}$. On the other hand $k_\beta(\mathbf{x}, \mathbf{y})$ itself is lower bounded in $\mathbf{x}$ and $\mathbf{y}$ under the same setting. Consequently, **A**($k_\gamma$, $\gamma$) holds true whenever $\Lambda$ is a bounded set and the birth intensity $\beta(\mathbf{x})$ is bounded in $\mathbf{x}$. The latter is in particular true if $\beta$ satisfies the setting discussed in the previous paragraph. For **B**($k_\gamma$, $\gamma$), note that a rough control consists in upper-bounding the squared norm of the difference in (7) by twice the sum of each squared norm. Then the same bounds as previously can be used to check **B**($k_\gamma$, $\gamma$) in the same setting.

**Example 2 (continued).** Assume that $k_\beta^m(\mathbf{x}, \mathbf{z}) = e^{-\vartheta'_m S_m(\mathbf{x}, \mathbf{z})}/a^m(\mathbf{x})$, where $\vartheta_m = (\vartheta_{m,m'})_{m' \in \mathbb{M}}$, $a^m(\mathbf{x}) = \int_\Lambda e^{-\vartheta'_m S_m(\mathbf{x}, \mathbf{z})} dz$ and $S_m$ is the vector

$$S_m(\mathbf{x}, \mathbf{z}) = \left( \sum_{\substack{x_i=(z_i, m_i) \in \mathbf{x} \\ m_i=m'}} \Phi_{m, m'}(z-z_i) \right)_{m' \in \mathbb{M}}.$$
This corresponds to a potential of the form $V(\mathbf{x}) = \sum_{i \neq j} \vartheta_{m_i, m_j} \Phi_{m_i, m_j}(z_i - z_j)$ in Example 2. The parameters of interest here are all the $\vartheta_{m,m'}$, $m, m' \in \mathbb{M}$. Assume that for all $m, m' \in \mathbb{M}$, $\Phi_{m,m'}$ is bounded by $L$. Then if we denote by $n_{m'}(\mathbf{x})$ the number of particles in $\mathbf{x}$ having label $m'$, we have

$$ \|S_m(\mathbf{x}, z)\|^2 \le \sum_{m' \in \mathbb{M}} L^2 n_{m'}^2(\mathbf{x}) \le L^2 n(\mathbf{x}) \sum_{m' \in \mathbb{M}} n_{m'}(\mathbf{x}) = L^2 n^2(\mathbf{x}). $$

Assume in addition that there is a maximal number of particles $n^*$ and that $\Lambda$ is a bounded set, then using the previous upper-bound, we may prove by elementary inequalities that both $\mathbf{A}(k_\gamma, \gamma)$ and $\mathbf{B}(k_\gamma, \gamma)$ are satisfied.

Concerning the mutation kernel, let us inspect Example 3.

**Example 3 (continued).** Remember that for this example we have

$$ k_{\tau}(\vartheta, \mathbf{x}, \mathbf{y}) = \frac{1}{n(\mathbf{x})} p_{m_i, m} $$

if there exists $x_i \in \mathbf{x}$ and $m \in \mathbb{M}$ such that $\mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m)$. The parameter of interest $\vartheta$ in this example correspond to the non-null entries $p_{m,m'}$ of the transition matrix. Denote by $p_*$ the minimal value of these non-null entries. We have $\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2 = 1/n(\mathbf{x})^2$, so that

$$ \int_E \frac{\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})} \mathbf{1}_{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\tau(\mathbf{x}, d\mathbf{y}) \\ = \int_E \frac{\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})^2} k_\tau(\vartheta, \mathbf{x}, \mathbf{y}) \mathbf{1}_{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\tau(\mathbf{x}, d\mathbf{y}) \\ \le \frac{1}{p_*^2} \int_E k_\tau(\vartheta, \mathbf{x}, \mathbf{y}) \nu_\tau(\mathbf{x}, d\mathbf{y}) = \frac{1}{p_*^2}. $$

Consequently $\mathbf{A}(k_\gamma, \gamma)$ (where $\gamma = \tau$) is satisfied in this setting whenever $\tau(\mathbf{x})$ is bounded in $\mathbf{x}$, which holds true in the setting discussed in the first paragraph of this section. On the other hand, since $\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})$ does not depend on $\vartheta$, $f_\vartheta^{k_\tau}(\mathbf{x}, \rho) = 0$ in (7), so that $\mathbf{B}(k_\gamma, \gamma)$ trivially holds.

Concerning the inter-jump motion and assumptions $\mathbf{A}_{\text{move}}$ and $\mathbf{B}_{\text{move}}$, we consider as an example the Langevin diffusion introduced in Example 4.

**Example 4 (continued).** Assume the same kind of parametrisation of the potential as in Example 2 above, that is $v_{i,n}(\mathbf{Z}, \mathbf{m}) = -\sum_{j \neq i} \vartheta_{m_i, m_j} \Phi_{m_i, m_j}(z_i - z_j)$ in (4), where the parameters of interest are again all the $\vartheta_{m,m'}$, $m, m' \in \mathbb{M}$.
This corresponds to a potential of the form $V(\mathbf{x}) = \sum_{i \neq j} \vartheta_{m_i, m_j} \Phi_{m_i, m_j}(z_i - z_j)$ in Example 2. The parameters of interest here are all the $\vartheta_{m,m'}$, $m, m' \in \mathbb{M}$. Assume that for all $m, m' \in \mathbb{M}$, $\Phi_{m,m'}$ is bounded by $L$. Then if we denote by $n_{m'}(\mathbf{x})$ the number of particles in $\mathbf{x}$ having label $m'$, we have

$$ \|S_m(\mathbf{x}, z)\|^2 \leq \sum_{m' \in \mathbb{M}} L^2 n_{m'}^2(\mathbf{x}) \leq L^2 n(\mathbf{x}) \sum_{m' \in \mathbb{M}} n_{m'}(\mathbf{x}) = L^2 n^2(\mathbf{x}). $$

Assume in addition that there is a maximal number of particles $n^*$ and that $\Lambda$ is a bounded set, then using the previous upper-bound, we may prove by elementary inequalities that both $\mathbf{A}(k_\gamma, \gamma)$ and $\mathbf{B}(k_\gamma, \gamma)$ are satisfied.

Concerning the mutation kernel, let us inspect Example 3.

**Example 3 (continued).** Remember that for this example we have

$$ k_{\tau}(\vartheta, \mathbf{x}, \mathbf{y}) = \frac{1}{n(\mathbf{x})} p_{m_i, m} $$

if there exists $x_i \in \mathbf{x}$ and $m \in \mathbb{M}$ such that $\mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m)$. The parameter of interest $\vartheta$ in this example correspond to the non-null entries $p_{m,m'}$ of the transition matrix. Denote by $p_*$ the minimal value of these non-null entries. We have $\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2 = 1/n(\mathbf{x})^2$, so that

$$ \int_E \frac{\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})} \mathbf{1}_{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\tau(\mathbf{x}, d\mathbf{y}) \\ = \int_E \frac{\|\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})\|^2}{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})^2} k_\tau(\vartheta, \mathbf{x}, \mathbf{y}) \mathbf{1}_{k_\tau(\vartheta, \mathbf{x}, \mathbf{y})>0} \nu_\tau(\mathbf{x}, d\mathbf{y}) \\ \leq \frac{1}{p_*^2} \int_E k_\tau(\vartheta, \mathbf{x}, \mathbf{y}) \nu_\tau(\mathbf{x}, d\mathbf{y}) = \frac{1}{p_*^2}. $$

Consequently $\mathbf{A}(k_\gamma, \gamma)$ (where $\gamma = \tau$) is satisfied in this setting whenever $\tau(\mathbf{x})$ is bounded in $\mathbf{x}$, which holds true in the setting discussed in the first paragraph of this section. On the other hand, since $\nabla_1 k_\tau(\vartheta, \mathbf{x}, \mathbf{y})$ does not depend on $\vartheta$, $f_{\vartheta}^{k_\tau}(\mathbf{x}, \rho) = 0$ in (7), so that $\mathbf{B}(k_\gamma, \gamma)$ trivially holds.

Concerning the inter-jump motion and assumptions $\mathbf{A}_{\text{move}}$ and $\mathbf{B}_{\text{move}}$, we consider as an example the Langevin diffusion introduced in Example 4.

**Example 4 (continued).** Assume the same kind of parametrisation of the potential as in Example 2 above, that is $v_{i,n}(\mathbf{Z}, \mathbf{m}) = -\sum_{j \neq i} \vartheta_{m_i, m_j} \Phi_{m_i, m_j}(z_i - z_j)$ in (4), where the parameters of interest are again all the $\vartheta_{m,m'}$, $m, m' \in \mathbb{M}$.
Remember that $\sigma_i(\mathbf{Z}, \mathbf{m}) = \sigma_{m_i}$ is known. Since the drift function is linear in the parameters, it is easily seen that $\mathbf{A}_{\text{move}}$ is satisfied if $\Phi_{m,m'}$ is bounded for any $\mathbf{m}, \mathbf{m}' \in \mathbb{M}$ and there is a maximal number of particles $n^*$. As to $\mathbf{B}_{\text{move}}$, we clearly have $f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho) = 0$ in (8), so that this condition holds trivially true.

# 4 Simulations and data analysis

## 4.1 Simulation study

We simulate in this section several trajectories of the BDMM process, for characteristics explained below, and we evaluate the quality of estimation of a parameter involved in the birth kernel. The chosen characteristics are motivated by the application that we will conduct in the next section, and are in line with previous studies by [6] and [15]. We present them below without precisely detailing each value.

The particles do not possess continuous marks, but can be of 6 different types, which correspond on one hand to 2 distinct particle natures (playing the role of Rab-11 type proteins and Langerin type proteins in the subsequent application), and on the other hand to 3 possible motion regimes: Brownian, sub-diffusive (according to an Ornstein-Uhlenbeck process), or super-diffusive (according to a Brownian motion with linear drift). Thus, the space $\mathbb{M}$ contains 6 different labels, but mutations will only be possible between motion regimes and not between the type of particles, see below.

We assume that the movement of each particle is independent of the others and corresponds to one of the three aforementioned regimes. Birth rates are constant and differ only according to the Langerin or Rab-11 type. Death rates (also distinct for Langerin and Rab-11) are proportional to the number of particles present. Mutations correspond to a change in movement regime: their intensity is also constant. Regarding transitions, we assume that the death kernel is uniform: each particle has the same probability of disappearing when a death event occurs. For mutations, we follow the setting of Example 3, i.e., these occur for any particle uniformly, according to a fixed transition matrix. Finally, for births, we assume that they generate an equiprobable Langerin or Rab-11 particle, with a diffusion regime respecting predetermined proportions. The Rab-11 type particles are then generated uniformly in $\Lambda$. For the Langerin type particles, we assume that given the
Remember that $\sigma_i(\mathbf{Z}, \mathbf{m}) = \sigma_{m_i}$ is known. Since the drift function is linear in the parameters, it is easily seen that $\mathbf{A}_{\text{move}}$ is satisfied if $\Phi_{m,m'}$ is bounded for any $\mathbf{m}, \mathbf{m}' \in \mathbb{M}$ and there is a maximal number of particles $n^*$. As to $\mathbf{B}_{\text{move}}$, we clearly have $f_{\vartheta}^{\text{move}}(\mathbf{x}, \rho) = 0$ in (8), so that this condition holds trivially true.

# 4 Simulations and data analysis

## 4.1 Simulation study

We simulate in this section several trajectories of the BDMM process, for characteristics explained below, and we evaluate the quality of estimation of a parameter involved in the birth kernel. The chosen characteristics are motivated by the application that we will conduct in the next section, and are in line with previous studies by [6] and [15]. We present them below without precisely detailing each value.

The particles do not possess continuous marks, but can be of 6 different types, which correspond on one hand to 2 distinct particle natures (playing the role of Rab-11 type proteins and Langerin type proteins in the subsequent application), and on the other hand to 3 possible motion regimes: Brownian, sub-diffusive (according to an Ornstein-Uhlenbeck process), or super-diffusive (according to a Brownian motion with linear drift). Thus, the space $\mathbb{M}$ contains 6 different labels, but mutations will only be possible between motion regimes and not between the type of particles, see below.

We assume that the movement of each particle is independent of the others and corresponds to one of the three aforementioned regimes. Birth rates are constant and differ only according to the Langerin or Rab-11 type. Death rates (also distinct for Langerin and Rab-11) are proportional to the number of particles present. Mutations correspond to a change in movement regime: their intensity is also constant. Regarding transitions, we assume that the death kernel is uniform: each particle has the same probability of disappearing when a death event occurs. For mutations, we follow the setting of Example 3, i.e., these occur for any particle uniformly, according to a fixed transition matrix. Finally, for births, we assume that they generate an equiprobable Langerin or Rab-11 particle, with a diffusion regime respecting predetermined proportions. The Rab-11 type particles are then generated uniformly in $\Lambda$. For the Langerin type particles, we assume that given the
<figure><img src="image_3.png" /><figcaption>Figure 1: Example of the density (20) for the birth of a new Langerin particle, given the Rab-11 particles already present (shown as red dots).</figcaption></figure>

pre-jump configuration $\mathbf{x} \in E$, they are generated according to the density $k_{\beta}^{L}(\mathbf{x}, \cdot)$ defined for all $z \in \Lambda$ by

$$k_{\beta}^{L}(\mathbf{x}, z) = \frac{p}{n_{R}(\mathbf{x})} \sum_{i=1}^{n_{R}(\mathbf{x})} \frac{1}{2\pi\sigma^2} \exp\left(\frac{-\|z - z_i^R\|^2}{2\sigma^2}\right) + \frac{(1-p)}{|\Lambda|} \mathbf{1}_{\Lambda}(z), \quad (20)$$

where $n_R(\mathbf{x})$ denotes the number of existing Rab-11 particles and $z_i^R$ their spatial coordinates. This density is parametrized by $p \in [0, 1]$ and $\sigma > 0$. It is a convex combination between a mixture of normal distributions centered around the existing Rab-11 particles, with variance $\sigma^2$, like in Example 1, and a uniform distribution over $\Lambda$. The closer the parameter $p$ is to 1, the more likely Langerin and Rab-11 particles will tend to be colocalized, and the smaller the standard deviation $\sigma$ is, the stronger the proximity between the two types of particles will be. An illustration of this density is shown in Figure 1, for a given configuration of Rab-11 particles represented by red dots.

In line with the application of the next section, we are not interested in the estimation of all features of the process, but only of the parameter $\vartheta = (p, \log \sigma)$ from the continuous time observation of a single trajectory of the process. We performed 500 simulations of trajectories for $p = 0.2$ and $\log \sigma =$
<figure><img src="image_2.png" /><figcaption>Figure 1: Example of the density (20) for the birth of a new Langerin particle, given the Rab-11 particles already present (shown as red dots).</figcaption></figure>

pre-jump configuration $\mathbf{x} \in E$, they are generated according to the density $k_{\beta}^{L}(\mathbf{x}, \cdot)$ defined for all $z \in \Lambda$ by

$$k_{\beta}^{L}(\mathbf{x}, z) = \frac{p}{n_{R}(\mathbf{x})} \sum_{i=1}^{n_{R}(\mathbf{x})} \frac{1}{2\pi\sigma^2} \exp\left(\frac{-\|z - z_i^R\|^2}{2\sigma^2}\right) + \frac{(1-p)}{|\Lambda|} \mathbf{1}_{\Lambda}(z), \quad (20)$$

where $n_R(\mathbf{x})$ denotes the number of existing Rab-11 particles and $z_i^R$ their spatial coordinates. This density is parametrized by $p \in [0, 1]$ and $\sigma > 0$. It is a convex combination between a mixture of normal distributions centered around the existing Rab-11 particles, with variance $\sigma^2$, like in Example 1, and a uniform distribution over $\Lambda$. The closer the parameter $p$ is to 1, the more likely Langerin and Rab-11 particles will tend to be colocalized, and the smaller the standard deviation $\sigma$ is, the stronger the proximity between the two types of particles will be. An illustration of this density is shown in Figure 1, for a given configuration of Rab-11 particles represented by red dots.

In line with the application of the next section, we are not interested in the estimation of all features of the process, but only of the parameter $\vartheta = (p, \log \sigma)$ from the continuous time observation of a single trajectory of the process. We performed 500 simulations of trajectories for $p = 0.2$ and $\log \sigma =$
<figure><img src="image_4.png" /><figcaption>Figure 2: Estimation results for $(p, \log \sigma) = (0.2, 1.34)$ by MLE, based on 500 simulations of the BDMM process specified in Section 4.1. In red: 95% Gaussian confidence ellipsoid, based on the bivariate Gaussian distribution centred at the mean of all estimations, and with covariance matrix the mean of all estimates of the asymptotic covariance matrix $J^{-1}(\vartheta)$.</figcaption></figure>

1.34, on a time interval similar to the data studied in Section 4.2. Figure 2 represents the set of all 500 maximum likelihood estimates of $\vartheta$ obtained for all 500 simulations, deduced from the likelihood established in Theorem 1. In red, the mean of the 95% confidence ellipsoid is plotted, more precisely it is the Gaussian confidence ellipsoid obtained from the bivariate Gaussian distribution centred at the mean of the estimates, and with covariance the mean of the 500 covariance matrices $J_N^{-1}(\hat{\vartheta}_N)$, obtained as explained at the end of Section 3.2. One can observe that most of the estimates provide a consistent value of the parameter and that the mean estimated confidence ellipsoid encodes the estimation uncertainty fairly well.

## 4.2 Data analysis

The dataset we consider comes from the observation by TIRF (Total Internal Reflection Fluorescence) microscopy of the intracellular traffic of some molecules near the membrane of a living cell [4]. This provides a video sequence showing two types of proteins observed simultaneously in the same
<figure><img src="image_3.png" /><figcaption>Figure 2: Estimation results for $(p, \log \sigma) = (0.2, 1.34)$ by MLE, based on 500 simulations of the BDMM process specified in Section 4.1. In red: 95% Gaussian confidence ellipsoid, based on the bivariate Gaussian distribution centred at the mean of all estimations, and with covariance matrix the mean of all estimates of the asymptotic covariance matrix $J^{-1}(\vartheta)$.</figcaption></figure>

1.34, on a time interval similar to the data studied in Section 4.2. Figure 2 represents the set of all 500 maximum likelihood estimates of $\vartheta$ obtained for all 500 simulations, deduced from the likelihood established in Theorem 1. In red, the mean of the 95% confidence ellipsoid is plotted, more precisely it is the Gaussian confidence ellipsoid obtained from the bivariate Gaussian distribution centred at the mean of the estimates, and with covariance the mean of the 500 covariance matrices $J_N^{-1}(\hat{\vartheta}_N)$, obtained as explained at the end of Section 3.2. One can observe that most of the estimates provide a consistent value of the parameter and that the mean estimated confidence ellipsoid encodes the estimation uncertainty fairly well.

## 4.2 Data analysis

The dataset we consider comes from the observation by TIRF (Total Internal Reflection Fluorescence) microscopy of the intracellular traffic of some molecules near the membrane of a living cell [4]. This provides a video sequence showing two types of proteins observed simultaneously in the same
<figure><img src="image_5.png" /><figcaption>Figure 3: Left: a frame from the raw video sequence analysed in Section 4.2, showing Langerin proteins. Right: superposition of all trajectories of Langerin proteins obtained after post-processing (i.e. segmentation and tracking) of the whole raw video. The colors indicate the estimated motion regime.</figcaption></figure>

cell: Langerin proteins and Rab-11 proteins. The image on the left of Figure 3 shows a frame obtained from the video of the Langerin proteins. After post-processing following [23], the proteins of interest are identified, represented by a point and tracked by the U-track algorithm [10] along the video sequence to provide trajectories, such as those visible in the right representation of Figure 3. These trajectories have been further analysed by the method developed in [5] to classify them into three diffusion regimes (these are the colors visible on the figure). The full dynamics is in line with a BDMM process: in addition to their displacement, some proteins disappear in the course of time, others appear, and finally some change their diffusion regime, which corresponds in our model to a mutation (even if it is not a mutation in the biological sense of the term).

An important biological question is to determine if Langerin proteins tend to appear close to existing Rab-11 proteins. This is the phenomenon of colocalization, indicating a strong interaction between these two types of proteins. To answer this question we estimate by maximum likelihood the birth kernel of the Langerin proteins, assuming that it is of the form (20), as in our simulation study. We obtained the estimate $(\hat{p}, \log \sigma) = (0.069, 1.1)$, where the unit of $\sigma$ is the pixel (the image being of size 250 × 283, each pixel representing an area of 160 × 160 nm²). Figure 4 shows the likelihood value for our dataset, with respect to $p$ and $\log \sigma$, along with the empirical
<figure><img src="image_4.png" /><figcaption>Figure 3: Left: a frame from the raw video sequence analysed in Section 4.2, showing Langerin proteins. Right: superposition of all trajectories of Langerin proteins obtained after post-processing (i.e. segmentation and tracking) of the whole raw video. The colors indicate the estimated motion regime.</figcaption></figure>

cell: Langerin proteins and Rab-11 proteins. The image on the left of Figure 3 shows a frame obtained from the video of the Langerin proteins. After post-processing following [23], the proteins of interest are identified, represented by a point and tracked by the U-track algorithm [10] along the video sequence to provide trajectories, such as those visible in the right representation of Figure 3. These trajectories have been further analysed by the method developed in [5] to classify them into three diffusion regimes (these are the colors visible on the figure). The full dynamics is in line with a BDMM process: in addition to their displacement, some proteins disappear in the course of time, others appear, and finally some change their diffusion regime, which corresponds in our model to a mutation (even if it is not a mutation in the biological sense of the term).

An important biological question is to determine if Langerin proteins tend to appear close to existing Rab-11 proteins. This is the phenomenon of colocalization, indicating a strong interaction between these two types of proteins. To answer this question we estimate by maximum likelihood the birth kernel of the Langerin proteins, assuming that it is of the form (20), as in our simulation study. We obtained the estimate $(\hat{p}, \log \sigma) = (0.069, 1.1)$, where the unit of $\sigma$ is the pixel (the image being of size 250 × 283, each pixel representing an area of 160 × 160 nm²). Figure 4 shows the likelihood value for our dataset, with respect to $p$ and $\log \sigma$, along with the empirical
<figure><img src="image_6.png" /><figcaption>Figure 4: Likelihood of the BDMM model for the considered dataset, as a function of the parameters $p$ and $\log \sigma$. The estimated 95% confidence ellipsoid, centred at the maximum likelihood, is overlaid in black.</figcaption></figure>

95% confidence ellipsoid around the maximum, obtained from the estimation of the matrix $J^{-1}(\vartheta)$ explained at the end of Section 3.2. This estimate suggests that, for this dataset, about 7% of Langerin proteins are colocalized with Rab-11 ones, this proportion being significantly positive in view of the confidence ellipsoid, and each colocalized Langerin protein appears (with high probability) within $3\hat{\sigma} \times 160 = 528$ nm of a Rab-11 protein, where $\hat{\sigma} = \exp(\widehat{\log \sigma})$.

# A Proof of Theorem 1

To simplify notation in the proof, we drop the dependence in $\vartheta$ of all terms. To get the likelihood, we view $(X_t)_{t \ge 0}$ as a jump-move process, a more general Markov process than the BDMM process. A jump-move process alternates motions and jumps in $E$, see [14]. It is defined through an intensity function $\alpha$, a jump kernel $K(x, \cdot)$ and a continuous Markov process $Y$ on $E$ that drives the motion of the process. For a BDMM, we have $\alpha(x) = \beta(x) + \delta(x) + \tau(x)$
<figure><img src="image_5.png" /><figcaption>Figure 4: Likelihood of the BDMM model for the considered dataset, as a function of the parameters $p$ and $\log \sigma$. The estimated 95% confidence ellipsoid, centred at the maximum likelihood, is overlaid in black.</figcaption></figure>

95% confidence ellipsoid around the maximum, obtained from the estimation of the matrix $J^{-1}(\vartheta)$ explained at the end of Section 3.2. This estimate suggests that, for this dataset, about 7% of Langerin proteins are colocalized with Rab-11 ones, this proportion being significantly positive in view of the confidence ellipsoid, and each colocalized Langerin protein appears (with high probability) within $3\hat{\sigma} \times 160 = 528$ nm of a Rab-11 protein, where $\hat{\sigma} = \exp(\widehat{\log \sigma})$.

# A Proof of Theorem 1

To simplify notation in the proof, we drop the dependence in $\vartheta$ of all terms. To get the likelihood, we view $(X_t)_{t \ge 0}$ as a jump-move process, a more general Markov process than the BDMM process. A jump-move process alternates motions and jumps in $E$, see [14]. It is defined through an intensity function $\alpha$, a jump kernel $K(x, \cdot)$ and a continuous Markov process $Y$ on $E$ that drives the motion of the process. For a BDMM, we have $\alpha(x) = \beta(x) + \delta(x) + \tau(x)$
and

$$K(\mathbf{x}, .) = \frac{\beta(\mathbf{x})}{\alpha(\mathbf{x})} K_{\beta}(\mathbf{x}, .) + \frac{\delta(\mathbf{x})}{\alpha(\mathbf{x})} K_{\delta}(\mathbf{x}, .) + \frac{\tau(\mathbf{x})}{\alpha(\mathbf{x})} K_{\tau}(\mathbf{x}, .). \quad (21)$$

By our assumptions

$$K(\mathbf{x}, d\mathbf{y}) = k(\mathbf{x}, \mathbf{y})\nu(\mathbf{x}, d\mathbf{y}), \quad (22)$$

where for any $F \in \mathcal{E}$, $\nu(\mathbf{x}, F) = \nu_{\beta}(\mathbf{x}, F)$ if there exist $A \subset \Lambda$ and $\mathcal{I} \subset \mathbb{M}$ such that $F = \mathbf{x} \cup (A \times \mathcal{I})$, $\nu(\mathbf{x}, F) = \nu_{\delta}(\mathbf{x}, F)$ if $\mathbf{x} \setminus x_i \in F$ for some $i$, and $\nu(\mathbf{x}, F) = \nu_{\tau}(\mathbf{x}, F)$ if there exist $m \in \mathbb{M}$ and $i$ such that $(\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m) \in F$. In turn, $k(\mathbf{x}, \mathbf{y}) = k_{\beta}(\mathbf{x}, \mathbf{y})\beta(\mathbf{x})/\alpha(\mathbf{x})$ if there exists $(z, m)$ such that $\mathbf{y} = \mathbf{x} \cup (z, m)$, $k(\mathbf{x}, \mathbf{y}) = k_{\delta}(\mathbf{x}, \mathbf{y})\delta(\mathbf{x})/\alpha(\mathbf{x})$ if $\mathbf{y} = \mathbf{x} \setminus x_i$ for some $i$, and $k(\mathbf{x}, \mathbf{y}) = k_{\tau}(\mathbf{x}, \mathbf{y})\tau(\mathbf{x})/\alpha(\mathbf{x})$ if there exist $m \in \mathbb{M}$ and $i$ such that $\mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m)$.

Note that a jump-move process observed continuously on $[0, t]$ has by definition a cadlag trajectory on $E$ and can equivalently be described by the vector $\underline{\mathbf{X}}$ defined by:

$$\underline{\mathbf{X}} = \sum_{n \in \mathbb{N}} (\mathbf{X}_0, \mathbf{X}_{[0,T_1]}, T_1, \mathbf{X}_{T_1}, \mathbf{X}_{[T_1,T_2]}, \dots, T_n - T_{n-1}, \mathbf{X}_{T_n}, \mathbf{X}_{[T_n,t]}) \mathbf{1}_{\{N_t=n\}}. \quad (23)$$

We set for any $i \ge 1$, $S_i = T_i - T_{i-1}$, $\mathbf{V}_i = \mathbf{X}_{T_i}$, and we recall that $(\mathbf{X}_t)_{T_i \le t < T_{i+1}} = (\mathbf{Y}_t^{(i)})_{0 \le t \le S_{i+1}}$, where $\mathbf{Y}^{(i)}$ is a continuous Markov process on $E$, identically distributed as $\mathbf{Y}$. With these notations $\underline{\mathbf{X}}$ is equivalent to $\tilde{\mathbf{X}}$ with:

$$\tilde{\underline{\mathbf{X}}} = \sum_{n \in \mathbb{N}} (\mathbf{V}_0, \mathbf{Y}_{[0,S_1]}^{(0)}, S_1, \mathbf{V}_1, \mathbf{Y}_{[0,S_2]}^{(1)}, \dots, S_n, \mathbf{V}_n, \mathbf{Y}_{[0,t-\sum_{j=1}^n S_j]}^{(n)}) \mathbf{1}_{\{N_t=n\}}. \quad (24)$$

Note that with this formalism, $\mathbf{Y}_0^{(i)} = \mathbf{V}_i$ for all $i$, and $\tilde{\underline{\mathbf{X}}}$ takes values in the space $\bigcup_{n \ge 0} \mathcal{W}_n$ where

$$\mathcal{W}_n = \bigotimes_{i=0}^{n} (E \times \mathcal{C}_{[S_i]} \times \mathbb{R}_+) \times E \times \mathcal{C}_{[t-\sum_{i=0}^{n} S_i]}$$

and where $\mathcal{C}_{[s]}$ denotes the space of continuous functions from $[0, s]$ to $E$.

As detailed in Section 2.3.3, the continuous Markov process $\mathbf{Y}_s^{(n)} = \Pi_n((\mathbf{Z}_s, \mathbf{m}))$ defined on $E_n$ and observed on $[0, t]$, given that $\mathbf{Z}_0 = \mathbf{z}$, admits the density $L(\mathbf{Z}_{[0,t]}, \mathbf{m})$ with respect to the reference process $\Pi_n((\mathbf{U}_t, \mathbf{m}))$ where
and

$$K(\mathbf{x}, .) = \frac{\beta(\mathbf{x})}{\alpha(\mathbf{x})} K_{\beta}(\mathbf{x}, .) + \frac{\delta(\mathbf{x})}{\alpha(\mathbf{x})} K_{\delta}(\mathbf{x}, .) + \frac{\tau(\mathbf{x})}{\alpha(\mathbf{x})} K_{\tau}(\mathbf{x}, .). \quad (21)$$

By our assumptions

$$K(\mathbf{x}, d\mathbf{y}) = k(\mathbf{x}, \mathbf{y})\nu(\mathbf{x}, d\mathbf{y}), \quad (22)$$

where for any $F \in \mathcal{E}$, $\nu(\mathbf{x}, F) = \nu_{\beta}(\mathbf{x}, F)$ if there exist $A \subset \Lambda$ and $\mathcal{I} \subset \mathbb{M}$ such that $F = \mathbf{x} \cup (A \times \mathcal{I})$, $\nu(\mathbf{x}, F) = \nu_{\delta}(\mathbf{x}, F)$ if $\mathbf{x} \setminus x_i \in F$ for some $i$, and $\nu(\mathbf{x}, F) = \nu_{\tau}(\mathbf{x}, F)$ if there exist $m \in \mathbb{M}$ and $i$ such that $(\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m) \in F$. In turn, $k(\mathbf{x}, \mathbf{y}) = k_{\beta}(\mathbf{x}, \mathbf{y})\beta(\mathbf{x})/\alpha(\mathbf{x})$ if there exists $(z, m)$ such that $\mathbf{y} = \mathbf{x} \cup (z, m)$, $k(\mathbf{x}, \mathbf{y}) = k_{\delta}(\mathbf{x}, \mathbf{y})\delta(\mathbf{x})/\alpha(\mathbf{x})$ if $\mathbf{y} = \mathbf{x} \setminus x_i$ for some $i$, and $k(\mathbf{x}, \mathbf{y}) = k_{\tau}(\mathbf{x}, \mathbf{y})\tau(\mathbf{x})/\alpha(\mathbf{x})$ if there exist $m \in \mathbb{M}$ and $i$ such that $\mathbf{y} = (\mathbf{x} \setminus (z_i, m_i)) \cup (z_i, m)$.

Note that a jump-move process observed continuously on $[0, t]$ has by definition a cadlag trajectory on $E$ and can equivalently be described by the vector $\underline{\mathbf{X}}$ defined by:

$$\underline{\mathbf{X}} = \sum_{n \in \mathbb{N}} (\mathbf{X}_{0}, \mathbf{X}_{[0, T_1]}, T_1, \mathbf{X}_{T_1}, \mathbf{X}_{[T_1, T_2]}, \dots, T_n - T_{n-1}, \mathbf{X}_{T_n}, \mathbf{X}_{[T_n, t]}) \mathbf{1}_{\{N_t=n\}}. \quad (23)$$

We set for any $i \ge 1$, $S_i = T_i - T_{i-1}$, $\mathbf{V}_i = \mathbf{X}_{T_i}$, and we recall that $(\mathbf{X}_t)_{T_i \le t < T_{i+1}} = (\mathbf{Y}_t^{(i)})_{0 \le t \le S_{i+1}}$, where $\mathbf{Y}^{(i)}$ is a continuous Markov process on $E$, identically distributed as $\mathbf{Y}$. With these notations $\underline{\mathbf{X}}$ is equivalent to $\tilde{\mathbf{X}}$ with:

$$\tilde{\underline{\mathbf{X}}} = \sum_{n \in \mathbb{N}} (\mathbf{V}_0, \mathbf{Y}_{[0, S_1]}^{(0)}, S_1, \mathbf{V}_1, \mathbf{Y}_{[0, S_2]}^{(1)}, \dots, S_n, \mathbf{V}_n, \mathbf{Y}_{[0, t-\sum_{j=1}^n S_j]}^{(n)}) \mathbf{1}_{\{N_t=n\}}. \quad (24)$$

Note that with this formalism, $\mathbf{Y}_0^{(i)} = \mathbf{V}_i$ for all $i$, and $\tilde{\underline{\mathbf{X}}}$ takes values in the space $\bigcup_{n \ge 0} \mathcal{W}_n$ where

$$\mathcal{W}_n = \bigotimes_{i=0}^{n} (E \times \mathcal{C}_{[S_i]} \times \mathbb{R}_+) \times E \times \mathcal{C}_{[t-\sum_{i=0}^{n} S_i]}$$

and where $\mathcal{C}_{[s]}$ denotes the space of continuous functions from $[0, s]$ to $E$.

As detailed in Section 2.3.3, the continuous Markov process $\mathbf{Y}_s^{(n)} = \Pi_n((\mathbf{Z}_s, \mathbf{m}))$ defined on $E_n$ and observed on $[0, t]$, given that $\mathbf{Z}_0 = \mathbf{z}$, admits the density $L(\mathbf{Z}_{[0,t]}, \mathbf{m})$ with respect to the reference process $\Pi_n((\mathbf{U}_t, \mathbf{m}))$ where
$\mathbf{U}_t = \mathbf{z} + \int_0^t \bar{\sigma}_n(\mathbf{U}_s, \mathbf{m}) dB_s$. Let us denote by $P_{[t]}^z$ the distribution of this reference process observed on $[0, t]$. For any measurable and bounded function $h$ defined on $E_n$, we thus have $\mathbb{E}(h(\mathbf{Y}_{[0,t]}^n)) = \int_{C_{[t]}} h(\mathbf{y}) L(\mathbf{y}_{[t]}) dP_{[t]}^{\mathbf{Y}_0^n}(\mathbf{y})$, where $L(\mathbf{y}_{[0,t]}, \mathbf{m})$ is a short notation for $L(\mathbf{z}_{[0,t]}, \mathbf{m})$ and where $\mathbf{y}_t = (\mathbf{z}_t, \mathbf{m})$ for any $t \ge 0$. Similarly, for $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ defined by $\mathbf{Y}_t = \sum_{n \ge 0} \mathbf{Y}_t^n \mathbf{1}_{\{\mathbf{Y}_0 \in E_n\}}$, we will write without ambiguity

$$ \mathbb{E}(h(\mathbf{Y}_{[0,t]})) = \int_{C_{[t]}} h(\mathbf{y}) L(\mathbf{y}_{[t]}) dP_{[t]}^{\mathbf{Y}_0}(\mathbf{y}), $$

where the cardinality $n$ defining $P_{[t]}^{\mathbf{Y}_0}$ is implicitly given by $n = n(\mathbf{Y}_0)$.

Let $h$ be a measurable and bounded function. For $n \in \mathbb{N}$, we use the conditional law of $S_{n+1}$ given $(\mathbf{X}_t)_{t \le T_n}$, $T_n$ and $\mathbf{Y}^{(n)}$, see (2), to compute:

$$
\begin{aligned}
& \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\
&= \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{S_{n+1} > t - \sum_{j=1}^n S_j\}} \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right] \\
&= \mathbb{E} \left[ \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{S_{n+1} > t - \sum_{j=1}^n S_j\}} \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right] \left| (\mathbf{X}_t)_{t \le T_n}, T_n, \mathbf{Y}^{(n)} \right] \right] \\
&= \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{Y}_u^{(n)}) du \right) \right] \\
&= \mathbb{E} \left[ \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{Y}_u^{(n)}) du \right) \right] \left| (\mathbf{X}_t)_{t \le T_n}, T_n \right] \right].
\end{aligned}
$$

Since given $S_1, \dots, S_n$ and $\mathbf{Y}_0^{(n)} = \mathbf{V}_n$, the Markov process $\mathbf{Y}^{(n)}$ observed on the time interval $[0, t - \sum_{j=1}^n S_j]$ admits the density $L(\mathbf{y}_{[t - \sum_{j=1}^n S_j]})$ with respect to $P_{[t - \sum_{j=1}^n S_j]}^{\mathbf{V}_n}$, we have

$$
\begin{aligned}
& \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\
&= \mathbb{E} \left[ \int_{C_{[t - \sum_{i=1}^n S_j]}} h(\mathbf{V}_0, \mathbf{Y}_{[0, S_1]}^{(0)}, \dots, S_n, \mathbf{V}_n, \mathbf{y}_{[t - \sum_{i=1}^n S_j]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right. \\
& \qquad \left. \times \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{y}_u^{(n)}) du \right) L(\mathbf{y}_{[t - \sum_{i=1}^n S_j]}^{(n)}) dP_{[t - \sum_{i=1}^n S_j]}^{\mathbf{V}_n}(\mathbf{y}^{(n)}) \right].
\end{aligned}
$$

Now, if we condition on $(\mathbf{X}_t)_{t \le T_{n-1}}$, $T_n$ and $\mathbf{Y}^{(n-1)}$ we know that the post-jump location $\mathbf{V}_n$ admits the density $k(\mathbf{X}_{T_n^{-}}, \mathbf{v}) = k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v})$ with respect
$\mathbf{U}_t = \mathbf{z} + \int_0^t \bar{\sigma}_n(\mathbf{U}_s, \mathbf{m}) dB_s$. Let us denote by $P_{[t]}^z$ the distribution of this reference process observed on $[0, t]$. For any measurable and bounded function $h$ defined on $E_n$, we thus have $\mathbb{E}(h(\mathbf{Y}_{[0,t]}^n)) = \int_{C_{[t]}} h(\mathbf{y}) L(\mathbf{y}_{[t]}) dP_{[t]}^{\mathbf{Y}_0^n}(\mathbf{y})$, where $L(\mathbf{y}_{[0,t]}, \mathbf{m})$ is a short notation for $L(\mathbf{z}_{[0,t]}, \mathbf{m})$ and where $\mathbf{y}_t = (\mathbf{z}_t, \mathbf{m})$ for any $t \ge 0$. Similarly, for $(\mathbf{Y}_t)_{t \ge 0}$ on $E$ defined by $\mathbf{Y}_t = \sum_{n \ge 0} \mathbf{Y}_t^n \mathbf{1}_{\{\mathbf{Y}_0 \in E_n\}}$, we will write without ambiguity

$$ \mathbb{E}(h(\mathbf{Y}_{[0,t]})) = \int_{C_{[t]}} h(\mathbf{y}) L(\mathbf{y}_{[t]}) dP_{[t]}^{\mathbf{Y}_0}(\mathbf{y}), $$

where the cardinality $n$ defining $P_{[t]}^{\mathbf{Y}_0}$ is implicitly given by $n = n(\mathbf{Y}_0)$.

Let $h$ be a measurable and bounded function. For $n \in \mathbb{N}$, we use the conditional law of $S_{n+1}$ given $(\mathbf{X}_t)_{t \le T_n}$, $T_n$ and $\mathbf{Y}^{(n)}$, see (2), to compute:

$$ \begin{aligned} & \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\ &= \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{S_{n+1} > t - \sum_{j=1}^n S_j\}} \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right] \\ &= \mathbb{E} \left[ \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{S_{n+1} > t - \sum_{j=1}^n S_j\}} \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right] | (\mathbf{X}_t)_{t \le T_n}, T_n, \mathbf{Y}^{(n)} \right] \\ &= \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{Y}_u^{(n)}) du \right) \right] \\ &= \mathbb{E} \left[ \mathbb{E} \left[ h(\tilde{\mathbf{X}}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{Y}_u^{(n)}) du \right) | (\mathbf{X}_t)_{t \le T_n}, T_n \right] \right]. \end{aligned} $$

Since given $S_1, \dots, S_n$ and $\mathbf{Y}_0^{(n)} = \mathbf{V}_n$, the Markov process $\mathbf{Y}^{(n)}$ observed on the time interval $[0, t - \sum_{j=1}^n S_j]$ admits the density $L(\mathbf{y}_{[t - \sum_{j=1}^n S_j]})$ with respect to $P_{[t - \sum_{j=1}^n S_j]}^{\mathbf{V}_n}$, we have

$$ \begin{aligned} & \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\ &= \mathbb{E} \left[ \int_{C_{[t - \sum_{i=1}^n S_j]}} h(\mathbf{V}_0, \mathbf{Y}_{[0, S_1]}^{(0)}, \dots, S_n, \mathbf{V}_n, \mathbf{y}_{[t - \sum_{i=1}^n S_j]}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right. \\ & \qquad \left. \times \exp \left( - \int_0^{t - \sum_{j=1}^n S_j} \alpha(\mathbf{y}_u^{(n)}) du \right) L(\mathbf{y}_{[t - \sum_{i=1}^n S_j]}^{(n)}) dP_{[t - \sum_{i=1}^n S_j]}^{\mathbf{V}_n}(\mathbf{y}^{(n)}) \right]. \end{aligned} $$

Now, if we condition on $(\mathbf{X}_t)_{t \le T_{n-1}}$, $T_n$ and $\mathbf{Y}^{(n-1)}$ we know that the post-jump location $\mathbf{V}_n$ admits the density $k(\mathbf{X}_{T_n^{-}}, \mathbf{v}) = k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v})$ with respect
to the measure $\nu(\mathbf{X}_{T_n^-, \cdot}) = \nu(\mathbf{Y}_{S_n}^{(n-1)}, \cdot)$ on $E$. Hence

$$ \mathbb{E}[h(\tilde{\mathbf{X}})\mathbf{1}_{\{N_t=n\}}] = \mathbb{E} \left[ \int_E \int_{C_{[t-\sum_{i=1}^n S_j]}} h(\mathbf{V}_0, \mathbf{Y}_{[0,S_1[}^{(0)}, \dots, S_n, \mathbf{v}_n, \mathbf{Y}_{[t-\sum_{i=1}^n S_j]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right. \\ \times \exp \left( -\int_0^{t-\sum_{j=1}^n S_j} \alpha(\mathbf{y}_u^{(n)}) du \right) L(\mathbf{y}_{[t-\sum_{i=1}^n S_j]}^{(n)}) dP_{[t-\sum_{i=1}^n S_j]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \\ \left. \times k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v}_n)\nu(\mathbf{Y}_{S_n}^{(n-1)}, d\mathbf{v}_n) \right]. $$

Conditioning on $(\mathbf{X}_t)_{t \le T_{n-1}}$, $T_{n-1}$ and $\mathbf{Y}^{(n-1)}$, we use again the law of $S_n$ given by (2) to get

$$ \mathbb{E}[h(\tilde{\mathbf{X}})\mathbf{1}_{\{N_t=n\}}] = \mathbb{E} \left[ \int_{\mathbb{R}_+} \int_E \int_{C_{[t-\sum_{i=1}^{n-1} S_j - s_n]}} h(\mathbf{V}_0, \mathbf{Y}_{[0,S_1[}^{(0)}, \dots, s_n, \mathbf{v}_n, \mathbf{Y}_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^{n-1} S_j + s_n \le t\}} \right. \\ \times \exp \left( -\int_0^{t-\sum_{j=1}^{n-1} S_j + s_n} \alpha(\mathbf{y}_u^{(n)}) du \right) L(\mathbf{y}_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{(n)}) dP_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \\ \left. \times k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v}_n)\nu(\mathbf{Y}_{S_n}^{(n-1)}, d\mathbf{v}_n)\alpha(\mathbf{Y}_{S_n}^{(n-1)}) \exp \left( -\int_0^{s_n} \alpha(\mathbf{Y}_u^{(n-1)}) du \right) ds_n \right]. $$

We can continue successively this process, using the (conditional) density of each move process $\mathbf{Y}^{(i)}$ on $[0, S_i]$, of each post-jump location $\mathbf{V}_i$, and of each
to the measure $\nu(\mathbf{X}_{T_n^-, \cdot}) = \nu(\mathbf{Y}_{S_n}^{(n-1)}, \cdot)$ on $E$. Hence

$$ \mathbb{E}[h(\tilde{\mathbf{X}})\mathbf{1}_{\{N_t=n\}}] = \mathbb{E}\left[ \int_E \int_{C_{[t-\sum_{i=1}^n S_j]}} h(\mathbf{V}_0, \mathbf{Y}_{[0,S_1[}^{(0)}, \dots, S_n, \mathbf{v}_n, \mathbf{Y}_{[t-\sum_{i=1}^n S_j]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^n S_j \le t\}} \right. \\ \left. \times \exp\left(-\int_0^{t-\sum_{j=1}^n S_j} \alpha(\mathbf{y}_u^{(n)}) du\right) L(\mathbf{y}_{[t-\sum_{i=1}^n S_j]}^{(n)}) dP_{[t-\sum_{i=1}^n S_j]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \right. \\ \left. \times k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v}_n)\nu(\mathbf{Y}_{S_n}^{(n-1)}, d\mathbf{v}_n) \right]. $$

Conditioning on $(\mathbf{X}_t)_{t \le T_{n-1}}$, $T_{n-1}$ and $\mathbf{Y}^{(n-1)}$, we use again the law of $S_n$ given by (2) to get

$$ \mathbb{E}[h(\tilde{\mathbf{X}})\mathbf{1}_{\{N_t=n\}}] = \mathbb{E}\left[ \int_{\mathbb{R}_+} \int_E \int_{C_{[t-\sum_{i=1}^{n-1} S_j - s_n]}} h(\mathbf{V}_0, \mathbf{Y}_{[0,S_1[}^{(0)}, \dots, s_n, \mathbf{v}_n, \mathbf{Y}_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^{n-1} S_j + s_n \le t\}} \right. \\ \left. \times \exp\left(-\int_0^{t-\sum_{j=1}^{n-1} S_j + s_n} \alpha(\mathbf{y}_u^{(n)}) du\right) L(\mathbf{y}_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{(n)}) dP_{[t-\sum_{i=1}^{n-1} S_j - s_n]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \right. \\ \left. \times k(\mathbf{Y}_{S_n}^{(n-1)}, \mathbf{v}_n)\nu(\mathbf{Y}_{S_n}^{(n-1)}, d\mathbf{v}_n)\alpha(\mathbf{Y}_{S_n}^{(n-1)}) \exp\left(-\int_0^{s_n} \alpha(\mathbf{Y}_u^{(n-1)}) du\right) ds_n \right]. $$

We can continue successively this process, using the (conditional) density of each move process $\mathbf{Y}^{(i)}$ on $[0, S_i]$, of each post-jump location $\mathbf{V}_i$, and of each
inter-jump time $S_i$, and we obtain

$$
\begin{align*}
& \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\
&= \int_{W_n} h(\mathbf{v}_0, \mathbf{y}_{[s_1]}^{(0)}, \dots, s_n, \mathbf{v}_n, \mathbf{y}_{[t-\sum_{i=1}^n s_j]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^n s_j \le t\}} \\
& \quad \times \exp \left( - \int_0^{t-\sum_{j=1}^n s_j} \alpha(\mathbf{y}_u^{(n)}) du \right) \times L(\mathbf{y}_{[t-\sum_{i=1}^n s_j]}^{(n)}) dP_{[t-\sum_{i=1}^n s_j]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \\
& \quad \times k(\mathbf{y}_{s_n}^{(n-1)}, \mathbf{v}_n) \nu(\mathbf{y}_{s_n}^{(n-1)}, d\mathbf{v}_n) \alpha(\mathbf{y}_{s_n}^{(n-1)}) \exp \left( - \int_0^{s_n} \alpha(\mathbf{y}_u^{(n-1)}) du \right) ds_n \\
& \quad \times \prod_{i=1}^{n-1} \left[ L(\mathbf{y}_{[s_{i+1}]}^{(i)}) dP_{[s_{i+1}]}^{\mathbf{v}_i}(\mathbf{y}^{(i)}) k(\mathbf{y}_{s_i}^{(i-1)}, \mathbf{v}_i) \nu(\mathbf{y}_{s_i}^{(i-1)}, d\mathbf{v}_i) \alpha(\mathbf{y}_{s_i}^{(i-1)}) \right] \\
& \quad \times \exp \left( - \int_0^{s_i} \alpha(\mathbf{y}_u^{(i-1)}) du \right) ds_i \right] L(\mathbf{y}_{[s_1]}^{(0)}) dP_{[s_1]}^{\mathbf{v}_0}(\mathbf{y}^{(0)}) p(d\mathbf{v}_0),
\end{align*}
$$

where $p$ denotes the distribution of the initial state on $E$. If we come back to the initial expression (23) of $\mathbf{X}$ from that of $\tilde{\mathbf{X}}$, using the shortcut $L(\mathbf{x}_{[t_i, t_{i+1}]}$ for $L(\mathbf{y}_{[s_{i+1}]}^{(i)}) = L(\mathbf{z}_{[0, t_{i+1}-t_i]}^{(i)}, \mathbf{m}^{(i)})$ in view of $\mathbf{x}_t = \{(\mathbf{z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ for all $t \in [t_i, t_{i+1}]$, we obtain

$$
\begin{align*}
\mathbb{E}[h(\underline{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] &= \int_{W_n} h(\mathbf{x}_0, \mathbf{x}_{[0,t_1]}, \dots, t_n - t_{n-1}, \mathbf{x}_{t_n}, \mathbf{x}_{[t_n,t]}) \mathbf{1}_{t_1 \le \dots \le t_n \le t} \\
&\quad \times \exp \left( - \int_0^t \alpha(\mathbf{x}_u) du \right) L(\mathbf{x}_{[t_n,t]}) \prod_{i=1}^n \left( L(\mathbf{x}_{[t_{i-1},t_i]}) k(\mathbf{x}_{t_i^-}, \mathbf{x}_{t_i}) \alpha(\mathbf{x}_{t_i^-}) \right) \eta^{(n)}(d\underline{\mathbf{x}}),
\end{align*}
$$

where $\eta^{(n)}$ is the measure given for $\underline{\mathbf{x}} = (\mathbf{x}_0, \mathbf{x}_{[0,t_1]}, \dots, t_n - t_{n-1}, \mathbf{x}_{t_n}, \mathbf{x}_{[t_n,t]})$ by

$$
\eta^{(n)}(d\underline{\mathbf{x}}) = \mathbf{1}_{t_1 \le \dots \le t_n \le t} dP_{[t_n, t]}^{\mathbf{x}_{t_n}}(\mathbf{x}) \nu(\mathbf{x}_{t_n^-}, d\mathbf{x}_{t_n}) dt_n \\
\times \prod_{i=1}^{n-1} \left( dP_{[t_i, t_{i+1}]}^{\mathbf{x}_{t_i}}(\mathbf{x}) \nu(\mathbf{x}_{t_i^-}, \mathbf{x}_{t_i}) dt_i \right) dP_{[0, t_1]}^{\mathbf{x}_0}(\mathbf{x}) p(d\mathbf{x}_0).
$$

Here, just as we have used the natural notation $L(\mathbf{x}_{[t_i, t_{i+1}]})$ for $L(\mathbf{y}_{[s_{i+1}]}^{(i)})$, we use $dP_{[t_i, t_{i+1}]}^{\mathbf{x}_{T_i}}(\mathbf{x})$ for $dP_{[s_{i+1}]}^{\mathbf{x}_{T_i}}(\mathbf{y}^{(i)})$.

Since for any $h$, $\mathbb{E}(h(\underline{\mathbf{X}})) = \sum_{n \ge 0} \mathbb{E}(h(\underline{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}})$, we have proven that the likelihood of $\mathbf{X}$ given by (23), with respect to the underlying measure
inter-jump time $S_i$, and we obtain

$$
\begin{align*}
& \mathbb{E}[h(\tilde{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] \\
&= \int_{W_n} h(\mathbf{v}_0, \mathbf{y}_{[s_1]}^{(0)}, \dots, s_n, \mathbf{v}_n, \mathbf{y}_{[t-\sum_{i=1}^n s_j]}^{(n)}) \mathbf{1}_{\{\sum_{j=1}^n s_j \le t\}} \\
&\quad \times \exp \left( -\int_0^{t-\sum_{j=1}^n s_j} \alpha(\mathbf{y}_u^{(n)}) du \right) \times L(\mathbf{y}_{[t-\sum_{i=1}^n s_j]}^{(n)}) dP_{[t-\sum_{i=1}^n s_j]}^{\mathbf{v}_n}(\mathbf{y}^{(n)}) \\
&\quad \times k(\mathbf{y}_{s_n}^{(n-1)}, \mathbf{v}_n) \nu(\mathbf{y}_{s_n}^{(n-1)}, d\mathbf{v}_n) \alpha(\mathbf{y}_{s_n}^{(n-1)}) \exp \left( -\int_0^{s_n} \alpha(\mathbf{y}_u^{(n-1)}) du \right) ds_n \\
&\quad \times \prod_{i=1}^{n-1} \left[ L(\mathbf{y}_{[s_{i+1}]}^{(i)}) dP_{[s_{i+1}]}^{\mathbf{v}_i}(\mathbf{y}^{(i)}) k(\mathbf{y}_{s_i}^{(i-1)}, \mathbf{v}_i) \nu(\mathbf{y}_{s_i}^{(i-1)}, d\mathbf{v}_i) \alpha(\mathbf{y}_{s_i}^{(i-1)}) \right] \\
&\quad \times \exp \left( -\int_0^{s_i} \alpha(\mathbf{y}_u^{(i-1)}) du \right) ds_i \right] L(\mathbf{y}_{[s_1]}^{(0)}) dP_{[s_1]}^{\mathbf{v}_0}(\mathbf{y}^{(0)}) p(d\mathbf{v}_0),
\end{align*}
$$

where $p$ denotes the distribution of the initial state on $E$. If we come back to the initial expression (23) of $\mathbf{X}$ from that of $\tilde{\mathbf{X}}$, using the shortcut $L(\mathbf{x}_{[t_i, t_{i+1}]}$ for $L(\mathbf{y}_{[s_{i+1}]}^{(i)}) = L(\mathbf{z}_{[0, t_{i+1}-t_i]}^{(i)}, \mathbf{m}^{(i)})$ in view of $\mathbf{x}_t = \{(\mathbf{z}_{t-T_i}^{(i)}, \mathbf{m}^{(i)})\}$ for all $t \in [t_i, t_{i+1}]$, we obtain

$$
\begin{align*}
\mathbb{E}[h(\underline{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}}] &= \int_{W_n} h(\mathbf{x}_0, \mathbf{x}_{[0,t_1]}, \dots, t_n - t_{n-1}, \mathbf{x}_{t_n}, \mathbf{x}_{[t_n,t]}) \mathbf{1}_{t_1 \le \dots \le t_n \le t} \\
&\quad \times \exp \left( -\int_0^t \alpha(\mathbf{x}_u) du \right) L(\mathbf{x}_{[t_n,t]}) \prod_{i=1}^n \left( L(\mathbf{x}_{[t_{i-1},t_i]}) k(\mathbf{x}_{t_i^-}, \mathbf{x}_{t_i}) \alpha(\mathbf{x}_{t_i^-}) \right) \eta^{(n)}(d\underline{\mathbf{x}}),
\end{align*}
$$

where $\eta^{(n)}$ is the measure given for $\underline{\mathbf{x}} = (\mathbf{x}_0, \mathbf{x}_{[0,t_1]}, \dots, t_n - t_{n-1}, \mathbf{x}_{t_n}, \mathbf{x}_{[t_n,t]})$ by

$$
\eta^{(n)}(d\underline{\mathbf{x}}) = \mathbf{1}_{t_1 \le \dots \le t_n \le t} dP_{[t_n, t]}^{\mathbf{x}_{t_n}}(\mathbf{x}) \nu(\mathbf{x}_{t_n^-}, d\mathbf{x}_{t_n}) dt_n \\
\times \prod_{i=1}^{n-1} \left( dP_{[t_i, t_{i+1}]}^{\mathbf{x}_{t_i}}(\mathbf{x}) \nu(\mathbf{x}_{t_i^-}, \mathbf{x}_{t_i}) dt_i \right) dP_{[0, t_1]}^{\mathbf{x}_0}(\mathbf{x}) p(d\mathbf{x}_0).
$$

Here, just as we have used the natural notation $L(\mathbf{x}_{[t_i, t_{i+1}]}$ for $L(\mathbf{y}_{[s_{i+1}]}^{(i)})$, we use $dP_{[t_i, t_{i+1}]}^{\mathbf{x}_{t_i}}(\mathbf{x})$ for $dP_{[s_{i+1}]}^{\mathbf{x}_{t_i}}(\mathbf{y}^{(i)})$.

Since for any $h$, $\mathbb{E}(h(\underline{\mathbf{X}})) = \sum_{n \ge 0} \mathbb{E}(h(\underline{\mathbf{X}}) \mathbf{1}_{\{N_t=n\}})$, we have proven that the likelihood of $\mathbf{X}$ given by (23), with respect to the underlying measure
$\eta(\cdot) = \sum_{n \in \mathbb{N}} \eta^{(n)}(\cdot \cap W_n)$ is given by

$$ \mathcal{L}(\underline{\mathbf{X}}) = \exp \left( - \int_{0}^{t} \alpha(\mathbf{X}_u) du \right) L(\mathbf{X}_{[t_{N_t}, t]}) \prod_{i=1}^{N_t} \left( L(\mathbf{X}_{[T_{i-1}, T_i]}) k(\mathbf{X}_{T_i^{-}}, \mathbf{X}_{T_i}) \alpha(\mathbf{X}_{T_i^{-}}) \right). $$

If we replace $\alpha$ and $k$ by their specific expression in (22), we obtain the result in the case of a BDMM process. Given that there is a one to one correspondance between the representation (23) and the trajectory $(\mathbf{X}_s)_{0 \le s \le t}$, we can view the above likelihood as the likelihood of $(\mathbf{X}_s)_{0 \le s \le t}$, where $\eta$ translates to a measure on the space of cadlag function from $[0, t]$ to $E$.

## B Proof of Theorem 2

Let us fix $N \ge 1$ and $h \in \mathbb{R}^l$. We let $\vartheta_N = \vartheta + h/\sqrt{N}$ and we denote

$$ \Lambda^{\vartheta_N/\vartheta}(t) = \log \frac{\mathcal{L}_t(\vartheta + \frac{h}{\sqrt{N}})}{\mathcal{L}_t(\vartheta)}. $$

By Theorem 1 the log-likelihood ratio process is given by:

$$ \Lambda^{\vartheta_N/\vartheta}(t) = \Lambda_L^{\vartheta_N/\vartheta}(t) + \Lambda_{\beta}^{\vartheta_N/\vartheta}(t) + \Lambda_{\delta}^{\vartheta_N/\vartheta}(t) + \Lambda_{\tau}^{\vartheta_N/\vartheta}(t) \\ + \Lambda_{k_{\beta}}^{\vartheta_N/\vartheta}(t) + \Lambda_{k_{\delta}}^{\vartheta_N/\vartheta}(t) + \Lambda_{k_{\tau}}^{\vartheta_N/\vartheta}(t), \quad (25) $$

where

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(t) = \int_{0}^{t} \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} \right) dN_s^{\gamma} - \int_{0}^{t} (\gamma(\vartheta_N, \mathbf{X}_s) - \gamma(\vartheta, \mathbf{X}_s)) ds, \quad (26) $$

$$ \Lambda_{k_{\gamma}}^{\vartheta_N/\vartheta}(t) = \int_{0}^{t} \log \left( \frac{k_{\gamma}(\vartheta_N, \mathbf{X}_{s^{-}}, \mathbf{X}_s)}{k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_s)} \right) dN_s^{\gamma}, \quad (27) $$

$$ \Lambda_L^{\vartheta_N/\vartheta}(t) = \log \frac{L(\vartheta_N, \mathbf{X}_{[T_{N_t}, t]})}{L(\vartheta, \mathbf{X}_{[T_{N_t}, t]})} + \sum_{i=0}^{N_t-1} \log \frac{L(\vartheta_N, \mathbf{X}_{[T_i, T_{i+1}[,})}{L(\vartheta, \mathbf{X}_{[T_i, T_{i+1}[,})}. \quad (28) $$

The proof of Theorem 2 follows a standard scheme as in [19] and [17]. The main step consists in proving the LAN decomposition for each of the three terms (26), (27) and (28) above, which is the statement of the three following lemmas. Then (17) is deduced by combining these decompositions, as carried out next.
$\eta(\cdot) = \sum_{n \in \mathbb{N}} \eta^{(n)}(\cdot \cap W_n)$ is given by

$$ \mathcal{L}(\underline{\mathbf{X}}) = \exp \left( - \int_{0}^{t} \alpha(\mathbf{X}_{u}) du \right) L(\mathbf{X}_{[t_{N_t}, t]}) \prod_{i=1}^{N_t} \left( L(\mathbf{X}_{[T_{i-1}, T_i]}) k(\mathbf{X}_{T_i^{-}}, \mathbf{X}_{T_i}) \alpha(\mathbf{X}_{T_i^{-}}) \right). $$

If we replace $\alpha$ and $k$ by their specific expression in (22), we obtain the result in the case of a BDMM process. Given that there is a one to one correspondance between the representation (23) and the trajectory $(\mathbf{X}_s)_{0 \le s \le t}$, we can view the above likelihood as the likelihood of $(\mathbf{X}_s)_{0 \le s \le t}$, where $\eta$ translates to a measure on the space of cadlag function from $[0, t]$ to $E$.

## B Proof of Theorem 2

Let us fix $N \ge 1$ and $h \in \mathbb{R}^l$. We let $\vartheta_N = \vartheta + h/\sqrt{N}$ and we denote

$$ \Lambda^{\vartheta_N/\vartheta}(t) = \log \frac{\mathcal{L}_t(\vartheta + \frac{h}{\sqrt{N}})}{\mathcal{L}_t(\vartheta)}. $$

By Theorem 1 the log-likelihood ratio process is given by:

$$ \Lambda^{\vartheta_N/\vartheta}(t) = \Lambda_L^{\vartheta_N/\vartheta}(t) + \Lambda_{\beta}^{\vartheta_N/\vartheta}(t) + \Lambda_{\delta}^{\vartheta_N/\vartheta}(t) + \Lambda_{\tau}^{\vartheta_N/\vartheta}(t) \\ + \Lambda_{k_{\beta}}^{\vartheta_N/\vartheta}(t) + \Lambda_{k_{\delta}}^{\vartheta_N/\vartheta}(t) + \Lambda_{k_{\tau}}^{\vartheta_N/\vartheta}(t), \quad (25) $$

where

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(t) = \int_{0}^{t} \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} \right) dN_s^{\gamma} - \int_{0}^{t} (\gamma(\vartheta_N, \mathbf{X}_s) - \gamma(\vartheta, \mathbf{X}_s)) ds, \quad (26) $$

$$ \Lambda_{k_{\gamma}}^{\vartheta_N/\vartheta}(t) = \int_{0}^{t} \log \left( \frac{k_{\gamma}(\vartheta_N, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})}{k_{\gamma}(\vartheta, \mathbf{X}_{s^{-}}, \mathbf{X}_{s})} \right) dN_s^{\gamma}, \quad (27) $$

$$ \Lambda_L^{\vartheta_N/\vartheta}(t) = \log \frac{L(\vartheta_N, \mathbf{X}_{[T_{N_t}, t]})}{L(\vartheta, \mathbf{X}_{[T_{N_t}, t]})} + \sum_{i=0}^{N_t-1} \log \frac{L(\vartheta_N, \mathbf{X}_{[T_i, T_{i+1}[,})}{L(\vartheta, \mathbf{X}_{[T_i, T_{i+1}[,})}. \quad (28) $$

The proof of Theorem 2 follows a standard scheme as in [19] and [17]. The main step consists in proving the LAN decomposition for each of the three terms (26), (27) and (28) above, which is the statement of the three following lemmas. Then (17) is deduced by combining these decompositions, as carried out next.
**Lemma 1.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) = h^T M_{\gamma}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{\gamma}^{N,\vartheta} \rangle(t)h + \mathrm{Rem}_{N,\vartheta,h}^{\gamma}(t), \quad (29) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{\gamma}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{\gamma}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (9).

**Lemma 2.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{k_{\gamma}}^{\vartheta_N/\vartheta}(tN) = h^T M_{k_{\gamma}}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{k_{\gamma}}^{N,\vartheta} \rangle(t)h + \mathrm{Rem}_{N,\vartheta,h}^{k_{\gamma}}(t), \quad (30) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{k_{\gamma}}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{k_{\gamma}}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (10).

**Lemma 3.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{L}^{\vartheta_N/\vartheta}(tN) = h^T M_{L}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{L}^{N,\vartheta} \rangle(t)h + \mathrm{Rem}_{N,\vartheta,h}^{L}(t), \quad (31) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{L}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{L}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (11).

The proofs of these three lemmas are postponed to the end of this section. Let us deduce (17). From their statements and (25), we have

$$ \Lambda^{\vartheta_N/\vartheta}(tN) = h^T M^{N,\vartheta}(t) - \frac{1}{2} h^T \sum_{i \in S} \langle M_i^{N,\vartheta} \rangle(t)h + \mathrm{Rem}_{N,\vartheta,h}(t), $$

where $S = \{L, \beta, \delta, \tau, k_{\beta}, k_{\delta}, k_{\tau}\}$, $M^{N,\vartheta}(t) = \sum_{i \in S} M_i^{N,\vartheta}$ and $\mathrm{Rem}_{N,\vartheta,h}(t) = \sum_{i \in S} \mathrm{Rem}_{N,\vartheta,h}^i(t)$. By Lemmas 1, 2 and 3, we know that $\mathrm{Rem}_{N,\vartheta,h}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability. To show the decomposition (17), it remains to prove that

$$ \langle M^{N,\vartheta} \rangle(t) = \sum_{i \in S} \langle M_i^{N,\vartheta} \rangle(t), $$

which, given the definition of $M^{N,\vartheta}(t)$, boils down to proving that

$$ \forall i, j \in S, i \neq j, \quad \langle M_i^{N,\vartheta}, M_j^{N,\vartheta} \rangle(t) = 0. \quad (32) $$
**Lemma 1.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) = h^T M_{\gamma}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{\gamma}^{N,\vartheta} \rangle(t) h + \mathrm{Rem}_{N,\vartheta,h}^{\gamma}(t), \quad (29) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{\gamma}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{\gamma}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (9).

**Lemma 2.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{k_{\gamma}}^{\vartheta_N/\vartheta}(tN) = h^T M_{k_{\gamma}}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{k_{\gamma}}^{N,\vartheta} \rangle(t) h + \mathrm{Rem}_{N,\vartheta,h}^{k_{\gamma}}(t), \quad (30) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{k_{\gamma}}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{k_{\gamma}}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (10).

**Lemma 3.** Under the same assumptions as in Theorem 2, we have for any $t > 0$, $h \in \mathbb{R}^l$ and $\vartheta \in \Theta$,

$$ \Lambda_{L}^{\vartheta_N/\vartheta}(tN) = h^T M_{L}^{N,\vartheta}(t) - \frac{1}{2} h^T \langle M_{L}^{N,\vartheta} \rangle(t) h + \mathrm{Rem}_{N,\vartheta,h}^{L}(t), \quad (31) $$

where $\mathrm{Rem}_{N,\vartheta,h}^{L}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability when $N \to +\infty$, and $(M_{L}^{N,\vartheta}(t))_{t>0}$ is a martingale with respect to $\mathcal{F}_{tN}$ given by (11).

The proofs of these three lemmas are postponed to the end of this section. Let us deduce (17). From their statements and (25), we have

$$ \Lambda^{\vartheta_N/\vartheta}(tN) = h^T M^{N,\vartheta}(t) - \frac{1}{2} h^T \sum_{i \in S} \langle M_i^{N,\vartheta} \rangle(t) h + \mathrm{Rem}_{N,\vartheta,h}(t), $$

where $S = \{L, \beta, \delta, \tau, k_{\beta}, k_{\delta}, k_{\tau}\}$, $M^{N,\vartheta}(t) = \sum_{i \in S} M_i^{N,\vartheta}$ and $\mathrm{Rem}_{N,\vartheta,h}(t) = \sum_{i \in S} \mathrm{Rem}_{N,\vartheta,h}^{i}(t)$. By Lemmas 1, 2 and 3, we know that $\mathrm{Rem}_{N,\vartheta,h}(t)$ tends to 0 in $\mathbb{P}_{x_0}^{\vartheta}$-probability. To show the decomposition (17), it remains to prove that

$$ \langle M^{N,\vartheta} \rangle(t) = \sum_{i \in S} \langle M_i^{N,\vartheta} \rangle(t), $$

which, given the definition of $M^{N,\vartheta}(t)$, boils down to proving that

$$ \forall i, j \in S, i \neq j, \quad \langle M_i^{N,\vartheta}, M_j^{N,\vartheta} \rangle(t) = 0. \quad (32) $$
First, for $i = L$, since $M_L^{N,\vartheta}(t)$ is a continuous martingale with finite variations, we have that $\langle M_L^{N,\vartheta}, M_j^{N,\vartheta} \rangle(t) = 0$ for any $j \neq L$, see [12]. Second, for all $i, j \in S \setminus \{L\}$, $i \neq j$, we have

$$[M_i^{N,\vartheta}, M_j^{N,\vartheta}](t) = \sum_{s \le t} \Delta M_i^{N,\vartheta}(s) \Delta M_j^{N,\vartheta}(s),$$

where $\Delta M_i^{N,\vartheta}(s) = M_i^{N,\vartheta}(s) - M_i^{N,\vartheta}(s^-)$. For $\gamma \neq \gamma'$, $M_\gamma^{N,\vartheta}(t)$ and $M_{\gamma'}^{N,\vartheta}(t)$ do not have any jump in common, and similarly for $M_\gamma^{N,\vartheta}(t)$ and $M_{k_\gamma'}^{N,\vartheta}(t)$, since a birth, a death or a mutation cannot occur at the same moment almost surely. Consequently $[M_\gamma^{N,\vartheta}, M_{\gamma'}^{N,\vartheta}](t) = [M_\gamma^{N,\vartheta}, M_{k_\gamma'}^{N,\vartheta}](t) = 0$ for any $\gamma \neq \gamma'$. This implies (32) for $i = \gamma$ and $j \in \{\gamma', k_\gamma'\}$, $\gamma \neq \gamma'$. To complete the proof of (32), it remains to show that $\langle M_\gamma^{N,\vartheta}, M_{k_\gamma'}^{N,\vartheta} \rangle(t) = 0$. We have that

$$[M_\gamma^{N,\vartheta}, M_{k_\gamma}^{N,\vartheta}](t) = \frac{1}{N} \int_0^{tN} \frac{\nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \frac{(\nabla_1 k_\gamma(\vartheta, \mathbf{X}_{s-}, \mathbf{X}_s))^T}{k_\gamma(\vartheta, \mathbf{X}_{s-}, \mathbf{X}_s)} dN_s^\gamma,$$

so that by Lemma 4, the associated angle bracket (corresponding to the compensator of the square bracket, see [12]) is given by

$$\langle M_\gamma^{N,\vartheta}, M_{k_\gamma}^{N,\vartheta} \rangle(t) = \frac{1}{N} \int_0^{tN} \nabla_1 \gamma(\vartheta, \mathbf{X}_s) \int_E (\nabla_1 k_\gamma(\vartheta, \mathbf{X}_s, \mathbf{y}))^T \nu_\gamma(\mathbf{X}_s, d\mathbf{y}) ds.$$

Remember that for any $\vartheta \in \Theta$ and any $\mathbf{x} \in E$, $\int_E k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})\nu(\mathbf{x}, d\mathbf{y}) = 1$. This implies $\int_E \nabla_1 k_\gamma(\vartheta, \mathbf{x}, \mathbf{y})\nu_\gamma(\mathbf{x}, d\mathbf{y}) = 0$ if interchange of integration and differentiation is valid. In particular, it can be showed that the latter holds under Condition **B**(k_γ, γ). Consequently, $\langle M_\gamma^{N,\vartheta}, M_{k_\gamma}^{N,\vartheta} \rangle(t) = 0$ and (32) is proven, which completes the proof of (17).

The last statement of Theorem 2, that is the convergence (18), is a direct consequence of Theorems 4.12 and 4.22 of [8], thanks to our hypothesis (H) that implies that $\emptyset$ is a recurrent atom for $(\mathbf{X}_t)_{t>0}$.

## B.1 Proof of Lemma 1

Let us prove that $Rem_{N,\vartheta,h}^\gamma(t)$ converges to 0 in $\mathbb{P}_{\mathbf{x}_0}^\vartheta$-probability, where

$$Rem_{N,\vartheta,h}^\gamma(t) = \Lambda_\gamma^{\vartheta_N/\vartheta}(tN) - h^T M_\gamma^{N,\vartheta}(t) + \frac{1}{2}h^T \langle M_\gamma^{N,\vartheta} \rangle(t)h,$$
First, for $i = L$, since $M_L^{N,\vartheta}(t)$ is a continuous martingale with finite variations, we have that $\langle M_L^{N,\vartheta}, M_j^{N,\vartheta} \rangle(t) = 0$ for any $j \neq L$, see [12]. Second, for all $i, j \in S \setminus \{L\}$, $i \neq j$, we have

$$[M_i^{N,\vartheta}, M_j^{N,\vartheta}](t) = \sum_{s \leq t} \Delta M_i^{N,\vartheta}(s) \Delta M_j^{N,\vartheta}(s),$$

where $\Delta M_i^{N,\vartheta}(s) = M_i^{N,\vartheta}(s) - M_i^{N,\vartheta}(s^-)$. For $\gamma \neq \gamma'$, $M_\gamma^{N,\vartheta}(t)$ and $M_{\gamma'}^{N,\vartheta}(t)$ do not have any jump in common, and similarly for $M_\gamma^{N,\vartheta}(t)$ and $M_{k_\gamma'}^{N,\vartheta}(t)$, since a birth, a death or a mutation cannot occur at the same moment almost surely. Consequently $[M_\gamma^{N,\vartheta}, M_{\gamma'}^{N,\vartheta}](t) = [M_\gamma^{N,\vartheta}, M_{k_\gamma'}^{N,\vartheta}](t) = 0$ for any $\gamma \neq \gamma'$. This implies (32) for $i = \gamma$ and $j \in \{\gamma', k_\gamma'\}$, $\gamma \neq \gamma'$. To complete the proof of (32), it remains to show that $\langle M_\gamma^{N,\vartheta}, M_{k_\gamma}^{N,\vartheta} \rangle(t) = 0$. We have that

$$[M_{\gamma}^{N,\vartheta}, M_{k_{\gamma}}^{N,\vartheta}](t) = \frac{1}{N} \int_{0}^{tN} \frac{\nabla_{1}\gamma(\vartheta, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \frac{(\nabla_{1}k_{\gamma}(\vartheta, \mathbf{X}_{s-}, \mathbf{X}_{s}))^{T}}{k_{\gamma}(\vartheta, \mathbf{X}_{s-}, \mathbf{X}_{s})} dN_{s^{\gamma}},$$

so that by Lemma 4, the associated angle bracket (corresponding to the compensator of the square bracket, see [12]) is given by

$$\langle M_{\gamma}^{N,\vartheta}, M_{k_{\gamma}}^{N,\vartheta} \rangle(t) = \frac{1}{N} \int_{0}^{tN} \nabla_{1}\gamma(\vartheta, \mathbf{X}_{s}) \int_{E} (\nabla_{1}k_{\gamma}(\vartheta, \mathbf{X}_{s}, \mathbf{y}))^{T} \nu_{\gamma}(\mathbf{X}_{s}, d\mathbf{y}) ds.$$

Remember that for any $\vartheta \in \Theta$ and any $\mathbf{x} \in E$, $\int_{E} k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})\nu(\mathbf{x}, d\mathbf{y}) = 1$. This implies $\int_{E} \nabla_{1}k_{\gamma}(\vartheta, \mathbf{x}, \mathbf{y})\nu_{\gamma}(\mathbf{x}, d\mathbf{y}) = 0$ if interchange of integration and differentiation is valid. In particular, it can be showed that the latter holds under Condition **B**(k_γ, γ). Consequently, $\langle M_\gamma^{N,\vartheta}, M_{k_\gamma}^{N,\vartheta} \rangle(t) = 0$ and (32) is proven, which completes the proof of (17).

The last statement of Theorem 2, that is the convergence (18), is a direct consequence of Theorems 4.12 and 4.22 of [8], thanks to our hypothesis (H) that implies that $\emptyset$ is a recurrent atom for $(X_t)_{t>0}$.

## B.1 Proof of Lemma 1

Let us prove that $Rem_{N,\vartheta,h}^{\gamma}(t)$ converges to 0 in $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}$-probability, where

$$Rem_{N,\vartheta,h}^{\gamma}(t) = \Lambda_{\gamma}^{\vartheta N/\vartheta}(tN) - h^T M_{\gamma}^{N,\vartheta}(t) + \frac{1}{2}h^T \langle M_{\gamma}^{N,\vartheta} \rangle(t)h,$$
and $M_{\gamma}^{N,\vartheta}(t)$ is given by (9). From (26), we get

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) = \int_{0}^{tN} \left[ \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} \right) - \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} + 1 \right] dN_{s}^{\gamma} + \int_{0}^{tN} \left[ \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} - 1 \right] dM_{s}^{\gamma}, $$

where $M_{t}^{\gamma} = N_{t}^{\gamma} - \int_{0}^{t} \gamma(\vartheta, \mathbf{X}_{s})ds$ is a $\mathcal{F}_{t}$-martingale. We have in particular $[M^{\gamma}]_{t} = N_{t}$ and $\langle M^{\gamma} \rangle_{t} = \int_{0}^{t} \gamma(\vartheta, \mathbf{X}_{s})ds$, see for instance [12]. We deduce that

$$ \begin{aligned} [M_{\gamma}^{N,\vartheta}](t) &= \frac{1}{N} \int_{0}^{tN} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma^{2}} (\vartheta, \mathbf{X}_{s^{-}}) d[M^{\gamma}]_{s} \\ &= \frac{1}{N} \int_{0}^{tN} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma^{2}} (\vartheta, \mathbf{X}_{s^{-}}) dN_{s}^{\gamma}. \end{aligned} $$

Note that $\gamma(\vartheta, \mathbf{X}_{T_{i}^{-}}) > 0$ for all $\gamma$-jump time $T_{i}$, so that the previous formula remains true with the addition of $\mathbf{1}_{\gamma(\vartheta, \mathbf{X}_{s^{-}})>0}$ in the integrand. So we get

$$ \langle M_{\gamma}^{N,\vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \mathbf{1}_{\gamma>0} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma} (\vartheta, \mathbf{X}_{s}) ds, $$

as claimed in (12), which well satisfies the property that $[M_{\gamma}^{N,\vartheta}](t) - \langle M_{\gamma}^{N,\vartheta} \rangle(t)$ is a martingale. Using these representations, we may write

$$ \begin{aligned} \operatorname{Rem}_{N,\vartheta,h}^{\gamma}(t) &= \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) - h^T M_{\gamma}^{N,\vartheta}(t) + \frac{1}{2} h^T (\langle M_{\gamma}^{N,\vartheta} \rangle(t) - [M_{\gamma}^{N,\vartheta}](t)) h + \frac{1}{2} h^T [M_{\gamma}^{N,\vartheta}](t) h \\ &= S_t^N + A_t^N + R_t^N, \end{aligned} $$

where

$$ \begin{aligned} S_t^N &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s^{-}})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s^{-}})} \right) dM_s^\gamma, \\ A_t^N &= \frac{1}{2N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) ds - \frac{1}{2N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s^{-}}) dN_s^\gamma, \\ R_t^N &= \int_0^{tN} \left( \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} \right) - \frac{\gamma(\vartheta_N, \mathbf{X}_{s^{-}})}{\gamma(\vartheta, \mathbf{X}_{s^{-}})} + 1 + \frac{1}{2N} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s^{-}}) \right) dN_s^\gamma. \end{aligned} $$
and $M_{\gamma}^{N, \vartheta}(t)$ is given by (9). From (26), we get

$$ \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) = \int_{0}^{tN} \left[ \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \right) - \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} + 1 \right] dN_{s}^{\gamma} + \int_{0}^{tN} \left[ \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right] dM_{s}^{\gamma}, $$

where $M_{t}^{\gamma} = N_{t}^{\gamma} - \int_{0}^{t} \gamma(\vartheta, \mathbf{X}_{s})ds$ is a $\mathcal{F}_{t}$-martingale. We have in particular $[M^{\gamma}]_{t} = N_{t}$ and $\langle M^{\gamma} \rangle_{t} = \int_{0}^{t} \gamma(\vartheta, \mathbf{X}_{s})ds$, see for instance [12]. We deduce that

$$ \begin{aligned} [M_{\gamma}^{N, \vartheta}](t) &= \frac{1}{N} \int_{0}^{tN} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma^{2}} (\vartheta, \mathbf{X}_{s-}) d[M^{\gamma}]_{s} \\ &= \frac{1}{N} \int_{0}^{tN} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma^{2}} (\vartheta, \mathbf{X}_{s-}) dN_{s}^{\gamma}. \end{aligned} $$

Note that $\gamma(\vartheta, \mathbf{X}_{T_{i}^{-}}) > 0$ for all $\gamma$-jump time $T_{i}$, so that the previous formula remains true with the addition of $\mathbf{1}_{\gamma(\vartheta, \mathbf{X}_{s-})>0}$ in the integrand. So we get

$$ \langle M_{\gamma}^{N, \vartheta} \rangle (t) = \frac{1}{N} \int_{0}^{tN} \mathbf{1}_{\gamma>0} \frac{(\nabla_{1}\gamma)(\nabla_{1}\gamma)^{T}}{\gamma} (\vartheta, \mathbf{X}_{s}) ds, $$

as claimed in (12), which well satisfies the property that $[M_{\gamma}^{N, \vartheta}](t) - \langle M_{\gamma}^{N, \vartheta} \rangle(t)$ is a martingale. Using these representations, we may write

$$ \begin{aligned} \operatorname{Rem}_{N, \vartheta, h}^{\gamma}(t) &= \Lambda_{\gamma}^{\vartheta_N/\vartheta}(tN) - h^T M_{\gamma}^{N, \vartheta}(t) + \frac{1}{2} h^T (\langle M_{\gamma}^{N, \vartheta} \rangle(t) - [M_{\gamma}^{N, \vartheta}](t)) h + \frac{1}{2} h^T [M_{\gamma}^{N, \vartheta}](t) h \\ &= S_t^N + A_t^N + R_t^N, \end{aligned} $$

where

$$ \begin{aligned} S_t^N &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})} \right) dM_s^\gamma, \\ A_t^N &= \frac{1}{2N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) ds - \frac{1}{2N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) dN_s^\gamma, \\ R_t^N &= \int_0^{tN} \left( \log \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \right) - \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} + 1 + \frac{1}{2N} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) \right) dN_s^\gamma. \end{aligned} $$
To complete the proof, we show below that each of these three terms tends to 0 in $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}$-probability.

First step: Convergence of $S_t^N$ to 0 in probability.

By the Markov inequality, we have for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta} (|S_t^N| > a) \le \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} ((S_t^N)^2) = \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} (\langle S^N \rangle_t), $$

where the last equality comes from the fact that $(S_t^N)^2 - \langle S^N \rangle_t$ is by definition a centred martingale (see [12]). We prove in the following that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta} (\langle S^N \rangle_t)$ tends to 0. We have

$$ \begin{aligned} [S^N]_t &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^-})}{\gamma(\vartheta, \mathbf{X}_{s^-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s^-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s^-})} \right)^2 dN_s^{\gamma} \\ &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^-})}{\gamma(\vartheta, \mathbf{X}_{s^-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s^-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s^-})} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_{s^-})>0} dN_s^{\gamma}, \end{aligned} $$

so that

$$ \langle S^N \rangle_t = \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_s)}{\gamma(\vartheta, \mathbf{X}_s)} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s)}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_s)} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_s)>0} \gamma(\vartheta, \mathbf{X}_s) ds. \quad (33) $$

Note that for any $\vartheta, \vartheta' \in \Theta$ and any $\mathbf{x}$,

$$ \gamma(\vartheta', \mathbf{x}) - \gamma(\vartheta, \mathbf{x}) = \int_0^1 (\nabla_1 \gamma(\vartheta + t(\vartheta' - \vartheta), \mathbf{x}))^T (\vartheta' - \vartheta) dt. $$

Thus we have for all $\mathbf{x} \in E$ and $\vartheta, \vartheta' \in \Theta$ such that $\gamma(\vartheta, \mathbf{x}) > 0$:

$$ \begin{aligned} & \left( \frac{\gamma(\vartheta', \mathbf{x})}{\gamma(\vartheta, \mathbf{x})} - 1 - \frac{(\nabla_1 \gamma(\vartheta, \mathbf{x}))^T (\vartheta' - \vartheta)}{\gamma(\vartheta, \mathbf{x})} \right)^2 \\ &= \left( \int_0^1 \frac{(\nabla_1 \gamma(\vartheta + t(\vartheta' - \vartheta), \mathbf{x}) - \nabla_1 \gamma(\vartheta, \mathbf{x}))^T (\vartheta' - \vartheta)}{\gamma(\vartheta, \mathbf{x})} dt \right)^2 \\ &\le \|\vartheta' - \vartheta\|^2 f_{\vartheta}^{\gamma}(\mathbf{x}, \|\vartheta' - \vartheta\|), \end{aligned} \quad (34) $$

where $f_{\vartheta}^{\gamma}(\mathbf{x}, \rho)$ is given by (6). Applying this inequality in (33) with $\vartheta' = \vartheta_N = \vartheta + h/\sqrt{N}$, we obtain

$$ \langle S^N \rangle_t \le \frac{\|h\|^2}{N} \int_0^{tN} f_{\vartheta}^{\gamma} \left( \mathbf{X}_s, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{X}_s) ds. \quad (35) $$
To complete the proof, we show below that each of these three terms tends to 0 in $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}$-probability.

First step: Convergence of $S_t^N$ to 0 in probability.

By the Markov inequality, we have for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta} (|S_t^N| > a) \le \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} ((S_t^N)^2) = \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} (\langle S^N \rangle_t), $$

where the last equality comes from the fact that $(S_t^N)^2 - \langle S^N \rangle_t$ is by definition a centred martingale (see [12]). We prove in the following that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta} (\langle S^N \rangle_t)$ tends to 0. We have

$$ \begin{aligned} [S^N]_t &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^-})}{\gamma(\vartheta, \mathbf{X}_{s^-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s^-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s^-})} \right)^2 dN_s^{\gamma} \\ &= \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s^-})}{\gamma(\vartheta, \mathbf{X}_{s^-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s^-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s^-})} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_{s^-})>0} dN_s^{\gamma}, \end{aligned} $$

so that

$$ \langle S^N \rangle_t = \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_s)}{\gamma(\vartheta, \mathbf{X}_s)} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s)}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_s)} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_s)>0} \gamma(\vartheta, \mathbf{X}_s) ds. \quad (33) $$

Note that for any $\vartheta, \vartheta' \in \Theta$ and any $\mathbf{x}$,

$$ \gamma(\vartheta', \mathbf{x}) - \gamma(\vartheta, \mathbf{x}) = \int_0^1 (\nabla_1 \gamma(\vartheta + t(\vartheta' - \vartheta), \mathbf{x}))^T (\vartheta' - \vartheta) dt. $$

Thus we have for all $\mathbf{x} \in E$ and $\vartheta, \vartheta' \in \Theta$ such that $\gamma(\vartheta, \mathbf{x}) > 0$:

$$ \begin{aligned} & \left( \frac{\gamma(\vartheta', \mathbf{x})}{\gamma(\vartheta, \mathbf{x})} - 1 - \frac{(\nabla_1 \gamma(\vartheta, \mathbf{x}))^T (\vartheta' - \vartheta)}{\gamma(\vartheta, \mathbf{x})} \right)^2 \\ &= \left( \int_0^1 \frac{(\nabla_1 \gamma(\vartheta + t(\vartheta' - \vartheta), \mathbf{x}) - \nabla_1 \gamma(\vartheta, \mathbf{x}))^T (\vartheta' - \vartheta)}{\gamma(\vartheta, \mathbf{x})} dt \right)^2 \\ &\le \|\vartheta' - \vartheta\|^2 f_{\vartheta}^{\gamma}(\mathbf{x}, \|\vartheta' - \vartheta\|), \end{aligned} \quad (34) $$

where $f_{\vartheta}^{\gamma}(\mathbf{x}, \rho)$ is given by (6). Applying this inequality in (33) with $\vartheta' = \vartheta_N = \vartheta + h/\sqrt{N}$, we obtain

$$ \langle S^N \rangle_t \le \frac{\|h\|^2}{N} \int_0^{tN} f_{\vartheta}^{\gamma} \left( \mathbf{X}_s, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{X}_s) ds. \quad (35) $$
Since $f_{\vartheta}^{\gamma}(\mathbf{x}, \cdot)$ is increasing, we have by condition **B**($\gamma$) that for any $\vartheta \in \Theta$ and $N$ large enough,

$$ \sup_{\mathbf{x} \in E} \left| f_{\vartheta}^{\gamma} \left( \mathbf{x}, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{x}) \right| < d(\vartheta), \qquad (36) $$

for some $d(\vartheta) > 0$. We can thus apply the ergodic assumption (**H**) to the right-hand side of (35) to get

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta} (|\langle S^N \rangle_t|) \le \frac{\|h\|^2 c(\mathbf{x}_0) d(\vartheta)}{rN} + \|h\|^2 t \int_E \left| f_{\vartheta}^{\gamma} \left( \mathbf{x}, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{x}) \right| \mu_{\infty}^{\vartheta}(d\mathbf{x}). $$

In view of (36), and since for any $\mathbf{x} \in E$, $f_{\vartheta}^{\gamma}(\mathbf{x}, \|h\|/\sqrt{N}) \to 0$ when $N \to 0$, we can conclude by the dominated convergence theorem.

**Second step: Convergence of $A_t^N$ to 0 in probability.**

Let $\epsilon > 0$. We have $2A_t^N = A_1 - A_2 - A_3$ where

$$ \begin{align} A_1 &= \frac{1}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_s) \right| > \epsilon \sqrt{N} \right\} ds, \notag \\ A_2 &= \frac{1}{N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_{s-}) \right| > \epsilon \sqrt{N} \right\} dN_s^{\gamma}, \tag{37} \\ A_3 &= \frac{1}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_{s-}) \right| \le \epsilon \sqrt{N} \right\} dM_s^{\gamma}. \notag \end{align} $$

Note that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(A_1) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(A_2) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_2|)$. We prove below that for any $\epsilon > 0$, $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \to 0$ as $N \to \infty$ and then we deal with the convergence in probability of $A_3$.

By condition **A**($\gamma$), we know that given $\vartheta \in \Theta$, the integrand in $A_1$ is uniformly bounded in $\mathbf{X}_s$ by some $d(\vartheta) > 0$. We can then apply (**H**) to get

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \le \frac{c(\mathbf{x}_0) d(\vartheta)}{N} + t \int_E \left| \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{x}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{x}) \right| > \epsilon \sqrt{N} \right\} \right| \mu_{\infty}^{\vartheta}(d\mathbf{x}). $$

This integrand is bounded thanks to condition **A**($\gamma$) and converges to 0 as $N \to \infty$ for any given $\mathbf{x}$ and $\vartheta$. We thus conclude by the dominated convergence theorem that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \to 0$ as $N \to 0$.
Since $f_{\vartheta}^{\gamma}(\mathbf{x}, \cdot)$ is increasing, we have by condition **B**($\gamma$) that for any $\vartheta \in \Theta$ and $N$ large enough,

$$ \sup_{\mathbf{x} \in E} \left| f_{\vartheta}^{\gamma} \left( \mathbf{x}, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{x}) \right| < d(\vartheta), \quad (36) $$

for some $d(\vartheta) > 0$. We can thus apply the ergodic assumption (**H**) to the right-hand side of (35) to get

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta} (|\langle S^N \rangle_t|) \le \frac{\|h\|^2 c(\mathbf{x}_0) d(\vartheta)}{rN} + \|h\|^2 t \int_E \left| f_{\vartheta}^{\gamma} \left( \mathbf{x}, \frac{\|h\|}{\sqrt{N}} \right) \gamma(\vartheta, \mathbf{x}) \right| \mu_{\infty}^{\vartheta}(d\mathbf{x}). $$

In view of (36), and since for any $\mathbf{x} \in E$, $f_{\vartheta}^{\gamma}(\mathbf{x}, \|h\|/\sqrt{N}) \to 0$ when $N \to 0$, we can conclude by the dominated convergence theorem.

**Second step: Convergence of $A_t^N$ to 0 in probability.**

Let $\epsilon > 0$. We have $2A_t^N = A_1 - A_2 - A_3$ where

$$ \begin{align} A_1 &= \frac{1}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_s) \right| > \epsilon \sqrt{N} \right\} ds, \notag \\ A_2 &= \frac{1}{N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_{s-}) \right| > \epsilon \sqrt{N} \right\} dN_s^{\gamma}, \tag{37} \\ A_3 &= \frac{1}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma^2} (\vartheta, \mathbf{X}_{s-}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{X}_{s-}) \right| \le \epsilon \sqrt{N} \right\} dM_s^{\gamma}. \notag \end{align} $$

Note that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(A_1) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(A_2) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_2|)$. We prove below that for any $\epsilon > 0$, $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \to 0$ as $N \to \infty$ and then we deal with the convergence in probability of $A_3$.

By condition **A**($\gamma$), we know that given $\vartheta \in \Theta$, the integrand in $A_1$ is uniformly bounded in $\mathbf{X}_s$ by some $d(\vartheta) > 0$. We can then apply (**H**) to get

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \le \frac{c(\mathbf{x}_0)d(\vartheta)}{N} + t \int_E \left| \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{x}) \mathbf{1} \left\{ \left| \frac{h^T \nabla_{1\gamma}}{\gamma} (\vartheta, \mathbf{x}) \right| > \epsilon \sqrt{N} \right\} \right| \mu_{\infty}^{\vartheta}(d\mathbf{x}). $$

This integrand is bounded thanks to condition **A**($\gamma$) and converges to 0 as $N \to \infty$ for any given $\mathbf{x}$ and $\vartheta$. We thus conclude by the dominated convergence theorem that $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) \to 0$ as $N \to 0$.
Concerning $A_3$, since it is a centred martingale, we have for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_3| > a) \le \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_3|^2) = \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle A_3 \rangle) $$

where

$$ \langle A_3 \rangle = \frac{1}{N^2} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^4}{\gamma^3} (\vartheta, \mathbf{X}_s) \mathbf{1} \left\{ \left| \frac{h^T \nabla_1 \gamma}{\gamma} (\vartheta, \mathbf{X}_s) \right| \le \epsilon \sqrt{N} \right\} ds \\ \le \frac{\epsilon^2}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) ds. $$

We can again apply (H) to the latest integral, thanks to condition A($\gamma$), and we obtain

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle A_3 \rangle) \le \epsilon^2 C(\mathbf{x}_0, \vartheta, h) $$

where $C(\mathbf{x}_0, \vartheta, h) > 0$. So for any $a > 0$ and any $\eta > 0$, we can choose $\epsilon$ so that $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_3| > a) \le \eta$. For these choices, we can further choose $N$ large enough so that $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_1| > a) \le \eta$ and $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_2| > a) \le \eta$, because for any $\epsilon > 0$, $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_2|) \to 0$ as $N \to \infty$. This entails the convergence of $A_t^N$ to 0 in $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}$-probability.

Third step: *Convergence of $R_t^N$ to 0 in probability.*

Denoting by $\varphi: x > -1 \mapsto \log(1+x) - x + x^2/2$, we have $R_t^N = R_1 - R_2/2$ with

$$ \begin{aligned} R_1 &= \int_0^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) dN_s^{\gamma}, \\ R_2 &= \int_0^{tN} \left( \left[ \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right]^2 - \left[ \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \right]^2 \right) dN_s^{\gamma}. \end{aligned} \quad (38) $$

We start by proving that $R_2$ tends to 0 in probability. Write $R_2 = \int (f^2 - g^2) dN_s^{\gamma}$ with obvious notations. We have by the Cauchy-Schwartz inequality and the fact that $(f+g)^2 = (f-g+2g)^2 \le 2(f-g)^2 + 8g^2$,

$$ \begin{aligned} R_2^2 &\le \int (f-g)^2 dN_s^{\gamma} \int (f+g)^2 dN_s^{\gamma} \\ &\le \left( \int (f-g)^2 dN_s^{\gamma} \right) \left( 8 \int (f-g)^2 dN_s^{\gamma} + 8 \int g^2 dN_s^{\gamma} \right). \end{aligned} $$
Concerning $A_3$, since it is a centred martingale, we have for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_3| > a) \le \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_3|^2) = \frac{1}{a^2} \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle A_3 \rangle) $$

where

$$ \begin{aligned} \langle A_3 \rangle &= \frac{1}{N^2} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^4}{\gamma^3} (\vartheta, \mathbf{X}_s) \mathbf{1} \left\{ \left| \frac{h^T \nabla_1 \gamma}{\gamma} (\vartheta, \mathbf{X}_s) \right| \le \epsilon \sqrt{N} \right\} ds \\ &\le \frac{\epsilon^2}{N} \int_0^{tN} \mathbf{1}_{\gamma>0} \frac{(h^T \nabla_1 \gamma)^2}{\gamma} (\vartheta, \mathbf{X}_s) ds. \end{aligned} $$

We can again apply (H) to the latest integral, thanks to condition A($\gamma$), and we obtain

$$ \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle A_3 \rangle) \le \epsilon^2 C(\mathbf{x}_0, \vartheta, h) $$

where $C(\mathbf{x}_0, \vartheta, h) > 0$. So for any $a > 0$ and any $\eta > 0$, we can choose $\epsilon$ so that $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_3| > a) \le \eta$. For these choices, we can further choose $N$ large enough so that $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_1| > a) \le \eta$ and $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}(|A_2| > a) \le \eta$, because for any $\epsilon > 0$, $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_1|) = \mathbb{E}_{\mathbf{x}_0}^{\vartheta}(|A_2|) \to 0$ as $N \to \infty$. This entails the convergence of $A_t^N$ to 0 in $\mathbb{P}_{\mathbf{x}_0}^{\vartheta}$-probability.

Third step: *Convergence of $R_t^N$ to 0 in probability.*

Denoting by $\varphi: x > -1 \mapsto \log(1+x) - x + x^2/2$, we have $R_t^N = R_1 - R_2/2$ with

$$ \begin{aligned} R_1 &= \int_0^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) dN_s^{\gamma}, \\ R_2 &= \int_0^{tN} \left( \left[ \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right]^2 - \left[ \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} \right]^2 \right) dN_s^{\gamma}. \end{aligned} \quad (38) $$

We start by proving that $R_2$ tends to 0 in probability. Write $R_2 = \int (f^2 - g^2) dN_s^{\gamma}$ with obvious notations. We have by the Cauchy-Schwartz inequality and the fact that $(f+g)^2 = (f-g+2g)^2 \le 2(f-g)^2 + 8g^2$,

$$ \begin{aligned} R_2^2 &\le \int (f-g)^2 dN_s^{\gamma} \int (f+g)^2 dN_s^{\gamma} \\ &\le \left( \int (f-g)^2 dN_s^{\gamma} \right) \left( 8 \int (f-g)^2 dN_s^{\gamma} + 8 \int g^2 dN_s^{\gamma} \right). \end{aligned} $$
To prove the convergence of $R_2$, it is then sufficient to show that $\int (f - g)^2 dN_s^\gamma = o_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$ and $\int g^2 dN_s^\gamma = O_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$. We have

$$ 
\begin{aligned}
& \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int (f-g)^2 dN_s^{\gamma} \\
&= \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})} \right)^2 dN_s^{\gamma} \quad (39) \\
&= \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_s)}{\gamma(\vartheta, \mathbf{X}_s)} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s)}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_s)} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_s)>0} \gamma(\vartheta, \mathbf{X}_s) ds,
\end{aligned}
$$

which is exactly $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle S^N \rangle_t)$, see (33). We have already proven that this term tends to 0, so $\int (f-g)^2 dN_s^\gamma = o_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$. Now, for any $\eta > 0$, by the Markov's inequality, we can choose $M > 0$ such that

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta} \left( \int g^2 dN_s^{\gamma} > M \right) \le \frac{1}{M} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \left( \frac{1}{N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s))^2}{\gamma(\vartheta, \mathbf{X}_s)} ds \right) \le \frac{t}{M} C(\vartheta, h) < \eta, $$

where $C(\vartheta, h)$ is a positive upper-bound deduced from condition A(γ). This proves that $\int g^2 dN_s^\gamma = O_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$ and completes the proof that $R_2$ tends to 0 in probability.

It remains to address the convergence of $R_1$. For any $\epsilon > 0$, we have $R_1 = R_{11} + R_{12}$ with

$$ R_{11} = \int_{0}^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma}, \quad (40) $$

$$ R_{12} = \int_{0}^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| \le \epsilon \right\}} dN_s^{\gamma}. \quad (41) $$

For $R_{11}$, note that for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta}(R_{11} > a) \le \mathbb{P}_{\mathbf{x}_0}^{\vartheta} \left( \int_0^{tN} \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma} > b \right), \quad (42) $$

for any $b < 1$. This is because when $b < 1$,

$$ \left( \int_0^{tN} \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma} \le b \right) \\ \implies \left( \text{for all } \gamma\text{-jump time } T_i \le tN, \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{T_i^{-}})}{\gamma(\vartheta, \mathbf{X}_{T_i^{-}})} - 1 \right| > \epsilon \right\}} = 0 \right) $$
To prove the convergence of $R_2$, it is then sufficient to show that $\int (f - g)^2 dN_s^\gamma = o_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$ and $\int g^2 dN_s^\gamma = O_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$. We have

$$ 
\begin{aligned}
& \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int (f - g)^2 dN_s^{\gamma} \\
& = \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})} \right)^2 dN_s^{\gamma} \quad (39) \\
& = \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_s)}{\gamma(\vartheta, \mathbf{X}_s)} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s)}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_s)} \right)^2 \mathbf{1}_{\gamma(\vartheta, \mathbf{X}_s)>0} \gamma(\vartheta, \mathbf{X}_s) ds,
\end{aligned}
$$

which is exactly $\mathbb{E}_{\mathbf{x}_0}^{\vartheta}(\langle S^N \rangle_t)$, see (33). We have already proven that this term tends to 0, so $\int (f-g)^2 dN_s^\gamma = o_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$. Now, for any $\eta > 0$, by the Markov's inequality, we can choose $M > 0$ such that

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta} \left( \int g^2 dN_s^{\gamma} > M \right) \le \frac{1}{M} \mathbb{E}_{\mathbf{x}_0}^{\vartheta} \left( \frac{1}{N} \int_0^{tN} \frac{(h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_s))^2}{\gamma(\vartheta, \mathbf{X}_s)} ds \right) \le \frac{t}{M} C(\vartheta, h) < \eta, $$

where $C(\vartheta, h)$ is a positive upper-bound deduced from condition A(γ). This proves that $\int g^2 dN_s^\gamma = O_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$ and completes the proof that $R_2$ tends to 0 in probability.

It remains to address the convergence of $R_1$. For any $\epsilon > 0$, we have $R_1 = R_{11} + R_{12}$ with

$$ R_{11} = \int_{0}^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma}, \quad (40) $$

$$ R_{12} = \int_{0}^{tN} \varphi \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right) \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| \le \epsilon \right\}} dN_s^{\gamma}. \quad (41) $$

For $R_{11}$, note that for any $a > 0$,

$$ \mathbb{P}_{\mathbf{x}_0}^{\vartheta}(R_{11} > a) \le \mathbb{P}_{\mathbf{x}_0}^{\vartheta} \left( \int_0^{tN} \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma} > b \right), \quad (42) $$

for any $b < 1$. This is because when $b < 1$,

$$ \left( \int_{0}^{tN} \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| > \epsilon \right\}} dN_s^{\gamma} \le b \right) \\ \implies \left( \text{for all } \gamma\text{-jump time } T_i \le tN, \mathbf{1}_{\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{T_i^{-}})}{\gamma(\vartheta, \mathbf{X}_{T_i^{-}})} - 1 \right| > \epsilon \right\}} = 0 \right) $$
which implies that $R_{11} = 0 \le a$. Moreover

$$ \mathbf{1}\{|\frac{\gamma(\vartheta_N, \mathbf{x})}{\gamma(\vartheta, \mathbf{x})} - 1| > \epsilon\} \le \mathbf{1}\{|\frac{\gamma(\vartheta_N, \mathbf{x})}{\gamma(\vartheta, \mathbf{x})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{x})}{\sqrt{N} \gamma(\vartheta, \mathbf{x})}| > \frac{\epsilon}{2}\} + \mathbf{1}\{|\frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{x})}{\sqrt{N} \gamma(\vartheta, \mathbf{x})}| > \frac{\epsilon}{2}\}. $$

Using this and the fact that for any $z$, $\mathbf{1}_{\{|z| > \frac{\epsilon}{2}\}} \le 4\epsilon^{-2}z^2\mathbf{1}_{\{|z| > \frac{\epsilon}{2}\}}$, we get

$$ \int_0^{tN} \mathbf{1}\left\{|\frac{\gamma(\vartheta_N, \mathbf{x}_{s-})}{\gamma(\vartheta, \mathbf{x}_{s-})} - 1\right| > \epsilon\right\} dN_s^\gamma \\ \le 4\epsilon^{-2} \int_0^{tN} \left(\frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 - \frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})}\right)^2 dN_s^\gamma \\ + 4\epsilon^{-2} \int_0^{tN} \left(\frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})}\right)^2 \mathbf{1}\left\{|\frac{h^T \nabla_1 \gamma(\vartheta, \mathbf{X}_{s-})}{\sqrt{N} \gamma(\vartheta, \mathbf{X}_{s-})}| > \frac{\epsilon}{2}\right\} dN_s^\gamma. $$

The first term is exactly $4\epsilon^{-2} \int (f-g)^2 dN_s^\gamma$, as already studied in (39), and the second term is $4\epsilon^{-2}A_2$ (up to $\epsilon/2$ instead of $\epsilon$), see (37). Both terms tend to 0 in $\mathbb{P}_{\mathbf{x}_0}^\vartheta$-probability whatever the value of $\epsilon > 0$. This proves that the right-hand side term in (42) tends to 0 for any $b < 1$, which yields the convergence of $R_{11}$ in $\mathbb{P}_{\mathbf{x}_0}^\vartheta$-probability to 0.

For $R_{12}$ given by (41), we choose $\epsilon < \frac{1}{2}$ and since $|\varphi(x)| \le 2|x|^3$ for $|x| \le \frac{1}{2}$, we have

$$ R_{12} \le \int_0^{tN} 2 \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right|^3 \mathbf{1}\left\{ \left| \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right| \le \epsilon \right\} dN_s^\gamma \\ \le 2\epsilon \int_0^{tN} \left( \frac{\gamma(\vartheta_N, \mathbf{X}_{s-})}{\gamma(\vartheta, \mathbf{X}_{s-})} - 1 \right)^2 dN_s^\gamma. $$

This last term is $2\epsilon(R_2 + \int g^2 dN_s^\gamma)$ if we use as previously the notation $R_2 = \int (f^2 - g^2)dN_s^\gamma$ for $R_2$ given by (38). We already know that $U := 2(R_2 + \int g^2 dN_s^\gamma)$ is a $O_{\mathbb{P}_{\mathbf{x}_0}^\vartheta}(1)$ as $N \to \infty$. This means that for any $\eta > 0$, there exists $M$ such that for $N$ sufficiently large $\mathbb{P}_{\mathbf{x}_0}^\vartheta(U > M) < \eta$. This implies that for any $a > 0$ and $\eta > 0$, we can choose $\epsilon$ small enough ($\epsilon < a/M$) so that $\mathbb{P}_{\mathbf{x}_0}^\vartheta(\epsilon U > a) < P_{\mathbf{x}_0}(U > M) < \eta$. So for any $a > 0$ and $\eta > 0$, we can choose $\epsilon$ so that $\mathbb{P}_{\mathbf{x}_0}^\vartheta(R_{12} > a) < \eta$ for $N$ large enough. The same inequality holds true for $R_{11}$ since we have proven that $R_{11}$ tends to 0 in probability for any $\epsilon > 0$. So $R_1 = R_{11} + R_{12}$ tends to 0 in $\mathbb{P}_{\mathbf{x}_0}^\vartheta$-probability, which concludes the proof.