# Maximum Likelihood Estimators for a Supercritical Branching Diffusion Process

Authors: Pablo Olivares; Janko Hernandez
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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.


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## Body

## 1. 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.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 Section2, 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.


## 2. Model and Main Notations

We establish the main features of our model.Starting from a fixed initial configuration, particles move independently inℝd 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.Let𝒰 be the set of all particles that can appear in the system; we represent 𝒰 by ⋃n=1∞‍ℕn.With every particleu∈𝒰 we associate a random vector (su,τu,Nu,(Xtu)t∈ℝ+) where su and τu are its birth and the death times, respectively, taking values on [0,∞], Xtu is its position at time t, and Nu represents the number of offsprings.At the initial timet=0, we have a configuration given by a finite number of particles denoted by u1,u2,…,uN∈𝒰 at respective deterministic positions x1,x2,…,xN. According to notations we establish
(2.1)su1=su2=⋯=suN=0,X0u1=x1,X0u2=x2,…,X0uN=xN.
We define recursively the random variables su, τu, and Xtu in the following way.Suppose a particlev dies, giving birth to a particle u among its descendants; we set su=τv. At time su, the particle u moves according to a diffusion process with drift b(·) and infinitesimal variance σ(·) then
(2.2)Xtu=Xsuu+∫0t‍1[su,∞)(s)b(Xsu)ds+∫0t‍1[su,∞)(s)σ(Xsu)·dWsufort≥0,
where Wu=(Wtu)t∈R+ is a standard Brownian motion in ℝd.The death rateλ(·) function, for a particle located at x at time t, satisfies
(2.3)Pr[τu≥t+Δt∣τu≥t≥su;Xtu=x]=λ(x)Δt+o(Δt).
Finally, the probability law representing the reproduction law of a particle located at point x, and denoted by (pk(x))k∈ℕ, x∈ℝd verifies
(2.4)Pr[Nu=k∣Xτuu=x]=pk(x).Processes(Wu)u∈𝒰 and (Nv)v∈𝒰 are independent.We describe the process of living particles by the measure-valued processM=(Mt)t≥0, where
(2.5)Mt=∑u∈𝒰‍1[su,τu)(t)δXtu.
Here δx denotes the Dirac measure on (ℝd,ℬ(ℝd)), where ℬ(ℝd) is the Borelian σ-algebra in ℝd.Notice that forA∈ℬ(ℝd), Mt(A) represents the number of living particles in the region A at time t.The processM is a Markov process called Branching Diffusion Process. For existence and properties see, for example, [5]. This process takes values in
(2.6)E={∑i=1n‍δxi:n=0,1,2,…;xi∈ℝd}
a closed subspace of 𝕄F(ℝd), the space of finite Borel positive measures on ℝd.Denote byCb(ℝd) the set of bounded and continuous functions on ℝd. For every f∈Cb(ℝd), we define the norm ∥f∥=sup{|f(x)|:x∈ℝd}.For everyξ∈𝕄F(ℝd) and f:ℝd→ℝ, measurable we set
(2.7)ξ(f)=∫ℝd‍fdξ.
We will note for 〈X,Y〉 and [X,Y] the covariance process and the quadratic covariance process. Also Xp 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;𝒜: class of processes in 𝒱 with EVar(A)∞≤∞, 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.Also𝒱loc, 𝒱loc+, 𝒜loc+, 𝒜loc, and ℳ(𝒫)loc are the corresponding local classes.We take(Mt)t∈R+ as the canonical process in the stochastic basis (Ω,ℱ,𝔽,Pm), where m is a given initial configuration, following its usual construction.By assuming that the functions driving the model depend on an unknown parameterθ, a statistical model associate to the process is considered.More specifically letΘ⊂ℝm 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)b:Θ×ℝd→ℝdb(θ;x)=bθ(x)=(biθ(x))i=1,…,d,σ:ℝd→ℝd⊗ℝdσ(x)=(σij(x))i,j=1,…,d,λ:Θ×ℝd→ℝ+*λ(θ;x)=λθ(x),p:Θ×ℕ×ℝd→[0,1]p(θ;k,x)=pk(θ;x)=pkθ(x).
Here ℝd⊗ℝd is the space of d×d real-valued matrices. When no confusion is possible we will note by |·| a norm in the space ℝd⊗ℝd as well as the Euclidean norm in ℝd. These functions define, for a given initial configuration m and any parameter θ, a probability Pmθ in the same way Pm is constructed.Suppose now these functions satisfy the following properties for everyθ∈Θ. (A1)
Lipshitz Local Condition. For all n≥1, there exists a constant Cnθ>0 such that
(2.9)|bθ(x)-bθ(y)|+|σ(x)-σ(y)|≤Cnθ|x-y|∀|x|≤n,|y|≤n.(A2)
Linear Growth Condition. There exists a constant K>0 nondepending on θ such that
(2.10)|bθ(x)|+|σ(x)|≤K(1+|x|)∀x∈ℝd.(A3)
σ(x) is an invertible matrix for all x∈ℝd, hence
(2.11)a(x)=σ(x)tσ(x)
is symmetric and positive definite.(A4)
For allx∈ℝd we have
(2.12)∑k=0∞‍pkθ(x)=1.(A5)
Let(2.13)mθ(x)=∑k=0∞‍kpkθ(x),κθ(x)=∑k=0∞‍(k-1)2pkθ(x)
then λθ, mθ, and κθ belong to Cb(ℝd) with
(2.14)λθ(x)≤∥λθ∥,|mθ(x)|≤∥mθ∥,|κθ(x)|≤∥κθ∥∀x∈ℝd.(A6)
There exist constantsλoθ>0 and moθ>1 such that
(2.15)λθ(x)≥λoθ,mθ(x)≥moθ>1∀x∈ℝd.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 onθ. 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)γ_θ=infx∈ℝdλθ(x)(mθ(x)-1),γ¯θ=supx∈ℝdλθ(x)(mθ(x)-1).
From (A5) and (A6) we have
(2.17)γ¯θ≤∥λθ∥(∥mθ∥-1),γ_θ≥λoθ(moθ-1)
then
(2.18)0<γ_θ≤λθ(x)(mθ(x)-1)≤γ¯θ<∞∀x∈ℝd.
The expression λθ(x)(mθ(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(2.19)ΔMt=Mt-Mt-,
where Mt- is the left limit of process (Mt)t≥0 at time t.Let’s denote by0<T1<T2<⋯<Tn<⋯ the times at which the jumps of the process take place, then, if at time Tn a particle dies at position Xn and has Kn offsprings we have
(2.20)ΔMTn=KnδXn-δXn.
The space of jumps is a closed subset of MF(ℝd) defines as
(2.21)Sd={(k-1)δx:k∈ℕ,x∈ℝd}.Let alsoμM be the random measure associated with the jumps of M given by
(2.22)μM(dt,dx)=∑s≤t‍1{ΔMs≠0}δ(s,ΔMs).
Finally, for every optional function W on ℝ+×Sd and a random measure ν on ℬ(ℝ+×Sd) we define the process W*ν by
(2.23)(W*ν)t=∫0t‍∫Sd-{0}‍W(s,x)ν(ds,dx).


## 3. Martingale Representation of the Process and Local Characteristics

We study now the local characteristics of the processM through the real process M(f)=(Mt(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 functionf:ℝd→ℝ in C2(ℝd), the process M(f) is decomposed as
(3.1)Mt(f)=M0(f)+∫0t‍Ms(Aθf)ds+Rft+idf*μMt,
where
(3.2)Rft=∑u∈𝒰‍∫0t‍1[su,τu)(s)(Dtf·σ)(Xsu)·dWsθ,u
is a square integrable martingale with zero mean under (Ω,ℱ,𝔽,Pmθ) and
(3.3)Aθf(x)=12∑i,j=1d‍aij(x)Di,jf(x)+∑i=1d‍biθ(x)Dif(x)
is the infinitesimal generator of the common diffusion law followed by the particles and idf:(t,(k-1)δx)↦(k-1)f(x) is optional on ℝ+×Sd.
HereDi and Di,j represent the first derivative with respect to xi and the mixed second derivative with respect to xi and xj, respectively, whereas D = (tD1,D2,…,Dd).Proof.
We apply Îto’s formula to process (2.2) for f∈C2(ℝd). Then we replace t by τu∧t and we get
(3.4)f(Xτu∧tu)-f(Xsuu)=∫0t1[su,τu)(s)Aθf(Xsu)ds+∫0t1[su,τu)(s){Dtf·σ}(Xsu)·dWsu.
Adding (3.4) for every u∈𝒰, the right hand side is
(3.5)∫0t{∑u∈𝒰‍1[su,τu)(s)Aθf(Xsu)}ds+∑u∈𝒰‍∫0t1[su,τu)(s){Dtf·σ}(Xsu)·dWsu=∫0tMs(Af)ds+Rft.
On the other hand, the left hand side can be written as
(3.6)∑u∈𝒰:su≤t‍{f(Xτu∧tu)-f(Xsuu)}=∑u∈𝒰:su≤t<τu‍f(Xtu)+∑u∈𝒰:τu≤t‍f(Xτuu)-∑u∈𝒰:su≤t‍f(Xsuu)=Mft-Mf0-∑0<s≤t‍ΔMfs=Mft-Mf0-idf*μMt.By definitionidf*(μM-νθ) is a local martingale, where νθ is the compensator of the process M then by adding and subtracting idf*νθ we have the following.Corollary 3.2.
For everyf∈C2(ℝd) the process
(3.7)Mt(f)-M0(f)-∫0t‍Ms(A*,θf)ds
is, under (Ω,ℱ,𝔽,Pmθ), a square integrable local martingale with zero mean and quadratic characteristic:
(3.8)∫0t‍Ms(Dtf·a·Df+λθκθf2)ds.
Here,
(3.9)A*,θf=Aθf+λθ(mθ-1)f.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.
LetX be a real-adapted process, h  a truncating function, A∈𝒱 continuous, C∈𝒱+ continuous, and ν a random measure in ℝ+×ℝd such that ν(dt,dy)=Kt(dy)dt. Let B=A+h*ν. ThenX is a semimartingale with local characteristics (B,C,ν) with respect to a truncating function h if and only if for every F∈C2(ℝ) the process
(3.10)F(Xt)-F(X0)-∫0tF′(Xs-)dAs-12∫0tF′′(Xs-)dCs-{F(Xs-+y)-F(Xs-)}*νt
is a local martingale.Proof.
It is enough to see thatB∈𝒱 is a continuous process and
(3.11)∫0tF(Xs-)d(h*ν)s=∫0tF(Xs-){∫ℝh(y)Ks(dy)}ds=∫(0,t]×ℝF(Xs-)h(y)ν(ds,dy)={F(Xs-)h(y)}*νt.
Also,(3.12)(y2∧1)*νt=∫0t{∫ℝ(y2∧1)Ks(dy)}ds=∫0tHsds,
where H is a nonnegative process then (y2∧1)*ν∈𝒱+. Moreover, it is continuous therefore predictable and it belongs to 𝒜loc+⊂𝒜loc.We have the following result.Theorem 3.4.
For anyθ∈Θ and m∈Ed there exist a probability Pmθ on as stochastic basis (Ω,ℱ,𝔽) such that (Ω,ℋ,ℍ,Pθ,m). We have M0=m a.s. and (Bf,Cf,νf) which are the local characteristics of M(f) with respect to h for any f∈Cb2(ℛd). The restriction Pθ,m to ℱ is the only probability in the filtered space (Ω,ℱ,𝔽) with these local characteristics. Here (Bf,Cf,νf) are given, for any truncating function h by
(3.13)Bft=∫0t‍Ms-(Aθf)(·)ds+h*νtf,Cft=∫0tMs-,t(Df(·)a(·)Df(·))ds
and νf on ℝ+×ℝd as
(3.14)νf(dt,dy)=Mt-(λ(·)∑k=0∞‍pkθ(·)δ(k-1)f(·)(dy))dt
or equivalently, for every optional function w on ℝ+×ℝd:
(3.15)w*νtf=∫0t‍Ms-(λ(·)∑k=0∞‍pk(·)w(s,(k-1)f(·)))ds.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 𝒢θ=𝒜θ+ℬθ where
(3.16)𝒜θF(μ(f))=F′(μ(f))μ(Aθf)+12F′′(μ(f))μ(DtfaDf),ℬθF(μ(f))=μ(λθ(·)∑k=0∞‍pkθ(·)[F(μ(f)+(k-1)δ.(f))-F(μ(f))]).
Moreover, for every non negative function F∈Cb2(ℝ) and f∈Cb2(ℝd) we have that
(3.17)F(Mt(f))-F(m(f))-∫0t𝒢θF(Ms-f)ds
is a local martingale with respect to (Ω,ℱ,𝔽,Pmθ).
We can write (3.17) as
(3.18)F(Mt(f))-F(M0(f))-∫0tF′(Ms-(f))Ms-(Aθf)ds-12∫0tF′′(Ms-)Ms-(Dtf·a·Df)ds-∫0tMs-(λθ∑k=0∞‍pkθ[Ff(Ms-+(k-1)δ.)-Ff(Ms-)])ds=F(Mtf)-F(M0f)-∫0tF′(Ms-f)d{∫0sMr-(Aθf)dr}-12∫0tF′′(Ms-f)d{∫0sMr-(Dtf·a·Df)dr}-{F(Ms-f+y)-F(Ms-f)}*νf(ds,dy)t.
From the last expression we apply the precedent proposition and identify the local characteristics as those in expressions (3.13) and (3.15).


## 4. Absolutely Continuous Measure Changes, Likelihood Function, and Fisher Information Measures

In this section, we calculate the likelihood function of the processMt based on a Girsanov theorem for semi-martingales.As reference measure we take the one determined by(4.1)b0(x)=0,λ0(x)=1,pk0(x)=12k+1,
that is, particles moving according to independent Brownian motions without drifts. In the sequel, as we start from a fix deterministic configuration M0=m, we will drop the dependence on m, then we denote by P0 and Pθ the respective probabilities generated by the reference measure and the functions given in (2.8) according to Theorem 3.4. We will denote by E0 and Eθ the expectations under P0 and Pθ, 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 measureQθ, absolutely continuous with respect to P0, with the same local characteristics than Pθ, therefore Qθ and Pθ are a.s. equal.Let(B0,C0,ν0) be the local characteristics of the process under P0 given by
(4.2)B0ft=∫0tMs-(A0f)ds+(hf*ν0)t,C0ft=∫0tMs-(DtfaDf)ds,(W*ν0)t=∫0tMs-(λ0(·)∑k=0∞‍pk0(·)W(s,(k-1)δ.))ds,
where
(4.3)A0f(x)=12∑i,j=1d‍Dijf(x)aij(x).
Equations (3.13) and (3.15) can be, respectively, rewritten as
(4.4)Bft=∫0tMs-(Aθf)ds+hf*νt,Cft=C0,W*νt=∫0tMs-(λθ∑k=0∞‍pkθW(s,(k-1)δ.))ds,
where
(4.5)Aθf=A0f+tDf·b.
Here δ. refers to the function x→δx.Next, we define the functiony:Sd→ℝ+ as
(4.6)y((k-1)δx)=λ(x)pkθ(x)λ0(x)pk0(x)
whenever λ0(x)pk0(x)≠0 and zero are otherwise.Also we define the following processes on(Ω,ℱ,𝔽):
(4.7)Yt=∑u∈𝒰‍∫0t1[su,τu)(bt·a-1)(Xsu)·dXsu+(y-1)*(μM-ν0)t,(4.8)Zt=exp(Yt-12〈Yc〉t)∏0<s≤t‍(1+ΔYs)e-ΔYs.
Note that Y is well defined on the basis (Ω,ℱ,𝔽,P0). Indeed,
(4.9){|y-1|*ν0}t=∫0tMs-(λ0∑k=0∞‍p0|λθpkθλ0pk0-1|)ds≤∫0tMs-(λ0∑k=0∞‍(λθpkθλ0+pk0))ds=∫0tMs-(λ0+λθ)ds≤∥λ0+λθ∥∫0tMs(1)ds
then |y-1|*ν0∈𝒜loc+(P0) and it is predictable.Moreover,(4.10)E0({|y-1|*ν0}t)≤∥λ0+λ∥E(∫0tMs(1)ds)<∞.
Then (y-1)*(μM-ν0) is a purely discontinuous local martingale on (Ω,ℱ,𝔽,P0).The first term inY is a local continuous martingale so the process Y is a local martingale. Their jumps have the form
(4.11)ΔYt=1[ΔMt≠0](y(ΔMt)-1)
then
(4.12)1+ΔYt={1ifΔMt=0,y(ΔMt)ifΔMt≠0.Hence,Z, the Doleans-Dole exponential local martingale of Y, is a local martingale on the same basis. Also Z≥0  P0-a.s. and E0Z0=1.Let(Rn)n∈ℕ be now a sequence of local stopping times for Z; we note by P0,Rn the restriction of P0 to the σ-algebra𝔽Rn and we define on it the probability measure Qn as dQn=ZRndP0,Rn, where ZRn is the process Z stopped at time Tn.We have the following result.Proposition 4.1.
The local characteristics ofQn are given by (3.13) and (3.15).Proof.
Let’s note by(Bn,Cn,νn) the local characteristics of M under the measure Qn.
First, note that ifW in ℝ+×Sd is an optional process then
(4.13)W*(yν0)t(ω)t=(Wy*ν0)t(ω)t=∫0tMs-(λ0∑k=0∞‍pk0W(s,(k-1)δ.)λθpkθλ0pk0)ds=∫0tMs-(λθ∑k=0∞‍pkθW(s,(k-1)δ.))ds=W*νt,
hence dν=ydν0 and in a similar way we have dνRn=ydν0Rn.
On the other hand,(4.14)Zt=exp(Yt--12〈Yc〉t)(∏0<s<t‍(1+ΔYs)e-ΔYs)(1+ΔYt)
then Z=Z-(1+ΔY) and
(4.15)1[ΔMtRn≠0]ZtRn=1[ΔMtRn≠0]Zt-Rny(ΔMtRn).
Thus, we have that for every 𝒫⊗𝒮-measurable function U:Ω×ℝ+×S→ℝ+d,
(4.16)∑t‍1[ΔMt≠0]ZtU(t,ΔMt)=∑t‍1[ΔMt≠0]y(ΔMt)Zt-U(t,ΔMt)
or equivalently
(4.17)ZU*μ∞X=yZ-U*μ∞X.
Hence,
(4.18)E(ZU*μ∞X)=E(yZ-U*μ∞X).
According to [10, Theorem III.3.17], yZ- is a version of the conditional expectation ℳμMP(Z∣𝒫⊗𝒮) and consequently yν0 is a version of the compensator of μX on the basis (Ω,ℱ,𝔽,Qθ). Then we have νn=ν.
Next, we defineN=Rf+hf*(μM-ν0) and N(n)=(N-[N,Y]p)Rn, where NRn is the process stopped at time Rn.
We can see thatN(n)∈ℳloc(Qn). Indeed, NRn∈ℳloc(P0) and its jumps
(4.19)ΔNRn=1[ΔMf≠0]hf(ΔMf)≤∥h∥
are bounded; hence combining Theorem III.3.11, Lemma III.3.14 in [10] we have that
(4.20)N(n)=(N-[N,Y]p)Rn∈ℳloc(Qn).
Moreover,
(4.21)[N,Y]=〈Rf,Yc〉+∑0<s≤t‍ΔNsΔYs=〈Rf,Yc〉+hf(y-1)*μM
and then
(4.22)N-[N,Y]p=Rf+hf*(μM-ν0)-〈Rf,Yc〉-hf*ν+hf*ν0=Rf-〈Rf,Yc〉+hf*(μM-ν).
We can write
(4.23)N(n)=(Rf-〈Rf,Yc〉)Rn+hf*(μM-ν)Rn∈ℳloc(Qn).
As(4.24)hf*(μM-ν)Rn=hf*(μMRn-νRn)∈ℳloc(Qn)
we have
(4.25)(Rf-〈Rf,Yc〉)Rn∈ℳloc(Qn).
From (3.1),
(4.26)MtRnf=M0Rnf+(∫0tMs(A0f)ds+Rf)Rn+id*μtMRnf=MRnf0+(∫0tMs(A0f)ds+〈Rf,Yc〉)Rn+(Rf-〈Rf,Yc〉)Rn+id*μtMRnf
with
(4.27)〈Rf,Yc〉=〈∑u∈𝒰‍∫0t1[su,τu)(s){Dtf·σ}(Xsu)·dWsu,∑u∈𝒰‍∫0t1[su,∞)(s)t(σ-1·bθ)(Xsu)·dWsu〉=∫0tMs(Dtf·bθ)ds.
We identify Bn and Cn as (3.13) using Proposition 3.3.From the previous proposition we get the following.Theorem 4.2.
Under conditions (A1)–(A6) in the space(Ω,ℱ,𝔽) we have for any θ∈Θ that Pθ≪locP0 with density Z given by (4.8).
The log-likelihood is given by(4.28)lt(θ)=∑u∈𝒰:su≤t‍∫suτu∧t{(tbθ·a-1)(Xsu)·dXsu-(12tbθ·a-1·bθ+λθ)(Xsu)ds}+∑n=1mt‍lnλθ(Xn)+∑n=1mt‍lnpNnθ(Xn),
where mt is the number of jumps before time t, and (Nn-1)δXn is the jump corresponding to time Tn.
HerePθ≪locP0 means that Pθ is locally absolutely continuous with respect to P0.Proof.
From Theorem3.4 we have the existence of the probability measure Pθ with local characteristics given by (3.13) and (3.15); by Proposition 4.1  Pθ and Qn are equal on the σ-algebra 𝔽Tn, therefore PTn≪(P0)Tn with density ZTn. By local uniqueness the result can be extended to the σ-algebra 𝔽.
From (4.8) we can write
(4.29)ln(Zt)=Yt-12〈Yc〉t+∑0<s≤t‍ln(1+ΔYs)-∑0<s≤t‍ΔYs=Ytc+(y-1)*(μM-ν0)t-12〈Yc〉t+∑0<s≤t‍ln(1+ΔYs)-(y-1)*μtM=Ytc-(y*ν0)t+(1*ν0)t-12〈Yc〉t+∑0<s≤t‍1[ΔMs≠0]y(ΔMs)
but
(4.30)(y*ν0)t=(1*yν0)t=1*νt=∫0tMs-(λθ∑k=0∞‍pkθ)ds=∑u∈𝒰‍∫0t1[su,τu)(s)λθ(Xsu)ds,〈Yc〉t=∑u∈𝒰‍∫0t1[su,τu)(s)(btθ·(σ-1)t)(Xsu)(σ-1·bθ)(Xsu)·d〈Wu〉s=∑u∈𝒰‍∫0t1[su,τu)(s)(btθ·a-1·bθ)(Xsu)ds.
Then
(4.31)ln(Zt)=∑u∈𝒰:su≤t‍∫suτu∧t{(btθ·a-1)(Xsu)·dXsu-(12tbθ·a-1·bθ+λθ)(Xsu)ds}+∑n=1mt‍{lnλθ(Xn)+lnpNnθ(Xn)}-∑n=1mt‍lnλ0pNn0(Xn)+(1*ν0)t.
Neglecting terms nondepending on θ we get (4.28).Next, we give expressions for the Fisher information and related measures. For details in the proofs and their asymptotic analysis we refer to [11].We denote by(l˙t(θ))t≥0 the score process, where the dot means the gradient with respect to the parameter θ. It is well known that (l˙t(θ))t≥0 is a zero mean martingale under Pθ. Its quadratic variation Jt(θ)=[l˙(θ)]t is the observed incremental information and the associate variance process It(θ)=〈l˙(θ)〉t is the expected incremental information. We denote by jt(θ)=-l¨t(θ) the Fisher observed information. Finally, the expected information is it(θ)=Eθ(l˙t(θ)l˙tt(θ)), see, for example, [15]. Among these four quantities we have the following relation:
(4.32)it(θ)=Eθ(Jt(θ))=Eθ(It(θ))=Eθ(jt(θ)).
We have the following result.Proposition 4.3.
If in addition to (A1)–(A6), we assume the following conditions:(B1)
The functionx↦σ-1(x) is bounded, that is, |σ-1(x)|≤∥σ-1∥<∞ for every x∈ℝd.(B2)
There exist constantsB1, B2, B3, Λ1, Λ2, P1, and P2, such that for every θ∈Θ, all x∈ℝd, all k∈ℕ, and all i,j,l=1,…,m, the following inequalities are satisfied:
(4.33)|Dibθ(x)|≤B1,|Dijbθ(x)|≤B2,|Dijlbθ(x)|≤B3,|Diλθ(x)|≤Λ1,|Dijλθ(x)|≤Λ2,|Dilnpkθ(x)|≤P1,|Dijlnpkθ(x)|≤P2,then J(θ),I(θ), and i(θ) are given by,
(4.34)Jt(θ)ij=∫0tMs(Dtibθa-1Djbθ)ds+{Di(lnλθ+lnpkθ)·Dj(lnλθ+lnpkθ)}*μtM,It(θ)ij=∫0tMs(ξijθ)ds,it(θ)ij=∑n=0M0(1)‍∫0t𝔼xnθ[ξijθ(Ys)exp{∫0s‍λθ(mθ-1)(Yr)dr}]ds,
where
(4.35)ξijθ=Dtibθa-1Djbθ+λθ·Dilnλθ·Djlnλθ+λθ·∑k=0∞‍Dilnpkθ·Djlnpkθ·pkθ.


## 5. A Branching Ornstein-Uhlenbeck Process

We consider a BDP where particles move according to an Ornstein-Uhlenbeck process onℝ, then
(5.1)Xt=φ∫0tXsds+Wt.
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 lawπ=(πk)k∈𝒩 satisfies
(5.2)π1=0,∑k=0∞‍kπk<∞,
where πk refers to the probability that a particle has k offsprings. Then the parameter is θ=(φ,λ,π)∈Θ where Θ⊂ℝ×(0,∞)×[0,1]ℕ.So we write(5.3)bθ(x)=φx,σθ(x)=1,λθ(x)=λ,pθ(x)=π.
From (4.28) we get
(5.4)lt(θ)=∑u∈𝒰:su≤t‍∫suτu∧t{φXsudXsu-12(φXsu)2ds-λds}+mtlnλ+∑n=1mt‍lnπNn=φ∑u∈𝒰:su≤t‍∫suτu∧tXsudXsu-φ22∑u∈𝒰:su≤t‍∫suτu∧t(Xsu)2ds-λ∑u∈𝒰:su≤t‍(τu∧t-su)+mtlnλ+∑n=1mt‍lnπNn.Noting that(5.5)∑u∈𝒰:su≤t‍(τu∧t-su)=K0T1+K1(T2-T1)+⋯+Kmt-1(Tmt-Tmt-1)+Kmt(t-Tmt),
where Kn is the number of particles alive on the interval [Tn,Tn+1) then
(5.6)Kn={N0ifn=0,N0+(N1-1)+⋯+(Nn-1)ifn>0,
where N0 is the number of ancestors and
(5.7)St=K0T1+K1(T2-T1)+⋯+Kmt-1(Tmt-Tmt-1)+Kmt(t-Tmt).
We finally have
(5.8)lt(θ)=φ2(∑u∈𝒰:su≤t‍{(Xτu∧tu)2-(Xsuu)2}-St)-φ22∑u∈𝒰:su≤t‍∫suτu∧t(Xsu)2ds-λSt+mtlnλ+∑n=1mt‍lnπNn.
From (5.8) we obtain the maximum likelihood estimators:
(5.9)φ^t=∑u∈𝒰:su≤t{(Xτu∧tu)2-(Xsuu)2}-St2∑u∈𝒰:su≤t∫suτu∧t(Xsu)2ds,λ^t=mtSt,π^n,t=rtnmt.
Here rtn is the number of splitting on (0,t) resulting in n offsprings.Moreover, we have the following results:(5.10)λ^t→a.s.λ,πn^t→a.s.πn∀n,mt(λλ^t-1)→ℒN(0,1),mtπn^t-πnπn(1-πn)→ℒN(0,1),∀n
suggesting consistency and asymptotic normality of the estimators in a more general context.We perform a simulation analysis for the model above in the following way.Equation (5.1) is discretized as
(5.11)Xt+h=Xt+φ∫tt+hXsds+(Wt+h-Wt).
For small h we take:
(5.12)Xt+h≈Xt+φhXt+ξh,
where ξh~N(0,h). As initial parameters we take
(5.13)φ=0.1,λ=0.05,π2=π3=12.Numerical results from simulated trajectories are shown in Table1. The particle system is observed until the time of the 1000th reproduction.Table 1
Parameter estimates of an Ornstein-Uhlenbeck process with two or three splitting. Five trajectories are simulated with parametersϕ=0.1, λ=0.05 and π2=π3=1/2.
φ
^
λ
^
π
2
^
π
3
^
0.10003
0.05139
0.499
0.501
0.10002
0.05152
0.525
0.475
0.10002
0.04974
0.524
0.476
0.10033
0.05028
0.498
0.502
0.10005
0.04852
0.524
0.476



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*Source: 515192-2012-11-20.xml*