Maximum Likelihood Estimators for a Supercritical Branching Diffusion Process

The log-likelihood of a nonhomogeneous Branching Diffusion Process under several conditions assuring existence and uniqueness of the diffusion part and nonexplosion of the branching process. Expressions for different Fisher information measures are provided. Using the semimartingale structure of the process and its local characteristics, a Girsanov-type result is applied. Finally, an Ornstein-Uhlenbeck process with finite reproduction mean is studied. Simulation results are discussed showing consistency and asymptotic normality.


Introduction
Some spatial-temporal models are often used to describe the behavior of particles, which are moving randomly in a domain and reproducing after a random time.
We consider a Branching Diffusion Process BDP , consisting in particles performing independent diffusion movements and having a random numbers of children at random times.In 1 , for example, a simple model of cells with binary splitting after an exponentially distributed random lifetime is considered, where cells move according independent Brownian motions.
More recently, 2 studied a model in order to describe pollution spread through dissemination of particles in the atmosphere.Additionally, the authors take into account the occurrence of particles' mass variations due to random divisions during their lifetimes.For applications in genetic populations see 3 .Also, in 4 , the recurrence of a BDP on manifolds is studied.

International Journal of Mathematics and Mathematical Sciences
In 5 , a particle system is considered in a more general context, where interaction among individuals is allowed.There, a link between the associated martingale problem and the infinitesimal generator is established.For a noninteracting BDP, the uniqueness of the martingale problem is found in 6 together with the analysis of the limit behavior of the process.
On the other hand, the statistical approach of this kind of models remains less explored.In 1 , under continuous observations upon a fixed time T , it obtained the maximum likelihood estimators for the variance and the rate of death of a Brownian motion with a deterministic binary reproduction law.In 7 , using a least square approach the parameters of the BDP are also estimated.
In 8, 9 , a birth and death processes in a flow particle system are considered.There, the absolute continuity of the probability law for the corresponding canonical process is obtained.We follow a similar approach, but allowing the possibility to have more than one particle at birth times, as our case, in which introduces additional complexity due to the exponential growth of the model.
There are many inference results for branching processes as well as for the diffusion process separately; we essentially consider both aspects together via a measure-valued process describing the particle configuration at any time.The functions describing the model i.e., drift, death rate, and reproduction law depend on a common unknown parameter.As in the model mentioned above, technical difficulties arise in writing the corresponding likelihood function.We use a Girsanov theorem for semimartingales, as given, for example, in 10 , allowing the passage from a BDP reference measure to another one depending on the true value of the parameter.The semi-martingale structure of the process and its corresponding local characteristics under the change of measure are obtained using Îto's formula.
The covariance matrix of the diffusion part is assumed to be known in order to avoid singularity with respect to the reference measure; otherwise the quadratic variation can be used as a nonparametric estimator of the former.
Expressions for the observed and expected Fisher information measures are provided.In a companion paper, see 11 , the asymptotic behavior of these measures is studied, and consequently, the consistency and asymptotic normality of the maximum likelihood estimators.
The organization of the paper is as follows.
In Section 2, we establish the model and the main notations.Also, we give certain sufficient conditions in order to have the existence of diffusion model and the nonexplosion on finite time of the branching part.These conditions are standards in both types of models.In Section 3, we obtain the semi-martingale structure of the model from Îto's formula and we calculate the local characteristics of the BDP.In Section 4, we find the likelihood function of the model using a Girsanov-type theorem for semi-martingales.Finally, in Section 5 we present an example, the Branching Ornstein-Uhlenbeck process, where explicit estimators can be obtained.

Model and Main Notations
We establish the main features of our model.
Starting from a fixed initial configuration, particles move independently in R d according to diffusion processes with the same drift and variance.Each particle dies after certain random time, depending on its trajectory.At the time of its death, it gives birth to an also random number of particles which continue to move from the ancestor position and reproduce in the same way.
Let U be the set of all particles that can appear in the system; we represent U by With every particle u ∈ U we associate a random vector s u , τ u , N u , X u t t∈R where s u and τ u are its birth and the death times, respectively, taking values on 0, ∞ , X u t is its position at time t, and N u represents the number of offsprings.
At the initial time t 0, we have a configuration given by a finite number of particles denoted by u 1 , u 2 , . . ., u N ∈ U at respective deterministic positions x 1 , x 2 , . . ., x N .According to notations we establish We define recursively the random variables s u , τ u , and X u t in the following way.Suppose a particle v dies, giving birth to a particle u among its descendants; we set s u τ v .At time s u , the particle u moves according to a diffusion process with drift b • and infinitesimal variance σ • then

International Journal of Mathematics and Mathematical Sciences
The process M is a Markov process called Branching Diffusion Process.For existence and properties see, for example, 5 .This process takes values in 2.7 We will note for X, Y and X, Y the covariance process and the quadratic covariance process.Also X p is the projection of X in the sense described in 10 .
We introduce the following spaces: V: class of right continuous-adapted processes with left limits and with finite variations on finite intervals starting at the origin at time 0; V : class of processes in V with nondecreasing trajectories; where Var A is the variation process associated to A; A : class of processes in V with EA ∞ ≤ ∞; M P : class of uniformly integrable martingales.
Also V loc , V loc , A loc , A loc , and M P loc are the corresponding local classes.
We take M t t∈R as the canonical process in the stochastic basis Ω, F, F, P m , where m is a given initial configuration, following its usual construction.
By assuming that the functions driving the model depend on an unknown parameter θ, a statistical model associate to the process is considered.
More specifically let Θ ⊂ R m be an open and convex set representing the parametric space and assume that b, λ, and p depend on a parameter θ ∈ Θ, then we have

2.8
Here R d ⊗ R d is the space of d × d real-valued matrices.When no confusion is possible we will note by | • | a norm in the space R d ⊗ R d as well as the Euclidean norm in R d .These functions define, for a given initial configuration m and any parameter θ, a probability P θ m in the same way P m is constructed.

2.15
Remark 2.1.A1 and A2 are standard conditions in order for the existence and uniqueness of the stochastic differential equations describing particle diffusions.
Remark 2.2.The infinitesimal covariance does not depend on θ.In general, we cannot have absolute continuity if σ depends on the parameter θ.This seems to be a constrain of the likelihood approach but in some cases it is possible to estimate σ using empirical quadratic covariations for example.
Remark 2.3.The second part of A6 is a uniform supercritical condition necessary to avoid the almost sure extinction of the branching process.
Let's now define

2.16
From A5 and A6 we have

2.18
The expression λ θ x m θ x − 1 is the generalized Malthus parameter, see, for example, 12, 13 .We assume that the whole process is observed on an interval 0, T ; that is, at every time we observe the entire configuration of particles.
We need to deal with the jumps of the process; to this end we define where M t− is the left limit of process M t t≥0 at time t.Let's denote by 0 • the times at which the jumps of the process take place, then, if at time T n a particle dies at position X n and has K n offsprings we have

2.20
The space of jumps is a closed subset of M F R d defines as

2.21
Let also μ M be the random measure associated with the jumps of M given by μ M dt, dx s≤t 1 {ΔM s / 0} δ s,ΔM s .

2.22
Finally, for every optional function W on R × S d and a random measure ν on B R × S d we define the process W * ν by W s, x ν ds, dx . 2.23

Martingale Representation of the Process and Local Characteristics
We study now the local characteristics of the process M through the real process M f M t f t≥0 .
The following result gives its semi-martingale structure, a useful decomposition of the process in a bounded variation process, a continuous martingale, and a purely discontinuous martingale.

Theorem 3.1. For every function
where is a square integrable martingale with zero mean under Ω, F, F, P θ m and is the infinitesimal generator of the common diffusion law followed by the particles and id f : Here D i and D i,j represent the first derivative with respect to x i and the mixed second derivative with respect to x i and x j , respectively, whereas Proof.We apply Îto's formula to process 2.2 for f ∈ C 2 R d .Then we replace t by τ u ∧ t and we get International Journal of Mathematics and Mathematical Sciences Adding 3.4 for every u ∈ U, the right hand side is

3.5
On the other hand, the left hand side can be written as

3.6
By definition id f * μ M − ν θ is a local martingale, where ν θ is the compensator of the process M then by adding and subtracting id f * ν θ we have the following.

Corollary 3.2. For every
is, under Ω, F, F, P θ m , a square integrable local martingale with zero mean and quadratic characteristic:

3.8
Here, Now, we calculate the local characteristics of the process 2.5 .We use the following result which is essentially a particular case of 10, Theorem II.2.42 see also 14 .

Proposition 3.3. Let X be a real-adapted process, h a truncating function, A ∈ V continuous, C ∈ V continuous, and ν a random measure in R × R d such that ν dt, dy K t dy dt. Let B A h * ν. Then X is a semimartingale with local characteristics B, C, ν with respect to a truncating function h if and only if for every
Proof.It is enough to see that B ∈ V is a continuous process and

3.11
Also, where H is a nonnegative process then y 2 ∧ 1 * ν ∈ V .Moreover, it is continuous therefore predictable and it belongs to A loc ⊂ A loc .
We have the following result.
Theorem 3.4.For any θ ∈ Θ and m ∈ E d there exist a probability P θ m on as stochastic basis Ω, F, F such that Ω, H, H, P θ,m .We have M 0 m a.s. and Bf, Cf, ν f which are the local characteristics of M f with respect to h for any f ∈ C 2 b R d .The restriction P θ,m to F is the only probability in the filtered space Ω, F, F with these local characteristics.Here Bf, Cf, ν f are given, for any truncating function h by International Journal of Mathematics and Mathematical Sciences or equivalently, for every optional function w on R × R d : Proof.From 5, Theorem 3.1 , or 6, Chapter 5 , we have the existence of a probability measure in Ω, F, F making 2.5 a BDP with infinitesimal generator G θ A θ B θ where

3.16
Moreover, for every non negative function is a local martingale with respect to Ω, F, F, P θ m .We can write 3.17 as

3.18
From the last expression we apply the precedent proposition and identify the local characteristics as those in expressions 3.13 and 3.15 .

Absolutely Continuous Measure Changes, Likelihood Function, and Fisher Information Measures
In this section, we calculate the likelihood function of the process M t based on a Girsanov theorem for semi-martingales.
As reference measure we take the one determined by b 0 x 0, 4.1 that is, particles moving according to independent Brownian motions without drifts.In the sequel, as we start from a fix deterministic configuration M 0 m, we will drop the dependence on m, then we denote by P 0 and P θ the respective probabilities generated by the reference measure and the functions given in 2.8 according to Theorem 3.4.We will denote by E 0 and E θ the expectations under P 0 and P θ , respectively.
It is well known that the semi-martingale structure persists after an absolutely continuous change of the probability measure.In order to see how the local characteristics change with it we construct a probability measure Q θ , absolutely continuous with respect to P 0 , with the same local characteristics than P θ , therefore Q θ and P θ are a.s.equal.
Let B 0 , C 0 , ν 0 be the local characteristics of the process under P 0 given by where Equations 3.13 and 3.15 can be, respectively, rewritten as where Here δ .refers to the function x → δ x .
Next, we define the function y : S d → R as whenever λ 0 x p 0 k x / 0 and zero are otherwise.Also we define the following processes on Ω, F, F : 1 ΔY s e −ΔY s .

4.8
Note that Y is well defined on the basis Ω, F, F, P 0 .Indeed, then |y − 1| * ν 0 ∈ A loc P 0 and it is predictable.Moreover,
The first term in Y is a local continuous martingale so the process Y is a local martingale.Their jumps have the form Hence, Z, the Doleans-Dole exponential local martingale of Y , is a local martingale on the same basis.Also Z ≥ 0 P 0 -a.s. and E 0 Z 0 1.
Let R n n∈N be now a sequence of local stopping times for Z; we note by P 0,R n the restriction of P 0 to the σ-algebra F R n and we define on it the probability measure Q n as dQ n Z R n dP 0,R n , where Z R n is the process Z stopped at time T n .
We have the following result.Proof.Let's note by B n , C n , ν n the local characteristics of M under the measure Q n .First, note that if W in R × S d is an optional process then 4.17 Hence,

4.18
According to 10, Theorem III.3.17 , yZ − is a version of the conditional expectation M P μ M Z | P ⊗ S and consequently yν 0 is a version of the compensator of μ X on the basis Ω, F, F, Q θ .Then we have ν n ν.
Next, we define N Rf h f * μ M − ν 0 and N n N − N, Y p R n , where N R n is the process stopped at time R n .

International Journal of Mathematics and Mathematical Sciences
We can see that N n ∈ M loc Q n .Indeed, N R n ∈ M loc P 0 and its jumps are bounded; hence combining Theorem III.3.11,Lemma III.3.14 in 10 we have that and then

4.22
We can write

4.23
As

4.27
We identify B n and C n as 3.13 using Proposition 3.3.
From the previous proposition we get the following.

4.28
where m t is the number of jumps before time t, and N n − 1 δ X n is the jump corresponding to time T n .
Here P θ loc P 0 means that P θ is locally absolutely continuous with respect to P 0 .
Proof.From Theorem 3.4 we have the existence of the probability measure P θ with local characteristics given by 3.13 and 3.15 ; by Proposition 4.1 P θ and Q n are equal on the σalgebra F T n , therefore P T n P 0 T n with density Z T n .By local uniqueness the result can be extended to the σ-algebra F.
From 4.8 we can write ln 1 ΔM s / 0 y ΔM s

4.29
International Journal of Mathematics and Mathematical Sciences but

4.30
Then 4.32 We have the following result.

A Branching Ornstein-Uhlenbeck Process
We consider a BDP where particles move according to an Ornstein-Uhlenbeck process on R, then X t ϕ t 0 X s ds W t .

5.1
The death rate λ ∈ 0, ∞ does not depend on the position; hence every particle has an exponential distributed lifetime independently of the trajectory.Its reproduction law π π k k∈N satisfies

5.13
Numerical results from simulated trajectories are shown in Table 1.The particle system is observed until the time of the 1000th reproduction.
the space of finite Borel positive measures on R d .Denote by C b R d the set of bounded and continuous functions on

Proposition 4 . 1 .
The local characteristics of Q n are given by 3.13 and 3.15 .
Here δ x denotes the Dirac measure onR d , B R d , where B R d is the Borelian σ-algebra in R d .Notice that for A ∈ B R d , M t A represents the number of living particles in the region A at time t.

Table 1 :
Parameter estimates of an Ornstein-Uhlenbeck process with two or three splitting.Five trajectories are simulated with parameters φ 0.1, λ 0.05 and π 2 π 3 1/2.asymptoticnormality of the estimators in a more general context.We perform a simulation analysis for the model above in the following way.
s ds W t h − W t .