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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 515192, 20 pages
Maximum Likelihood Estimators for a Supercritical Branching Diffusion Process
1Department of Mathematics, Ryerson University, Toronto, ON, Canada M5B 2K3
2School of Business, ITAM, 01000 Mexico City, DF, Mexico
Received 29 June 2012; Accepted 18 October 2012
Academic Editor: Angelo Plastino
Copyright © 2012 Pablo Olivares and Janko Hernandez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , 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,  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 . Also, in , the recurrence of a BDP on manifolds is studied.
In , 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  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 , under continuous observations upon a fixed time , 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 , 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 , 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 , 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.
2. Model and Main Notations
We establish the main features of our model.
Starting from a fixed initial configuration, particles move independently in 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 be the set of all particles that can appear in the system; we represent by .
With every particle we associate a random vector where and are its birth and the death times, respectively, taking values on , is its position at time , and represents the number of offsprings.
At the initial time , we have a configuration given by a finite number of particles denoted by at respective deterministic positions . According to notations we establish We define recursively the random variables , , and in the following way.
Suppose a particle dies, giving birth to a particle among its descendants; we set . At time , the particle moves according to a diffusion process with drift and infinitesimal variance then where is a standard Brownian motion in .
The death rate function, for a particle located at at time , satisfies Finally, the probability law representing the reproduction law of a particle located at point , and denoted by , verifies
Processes and are independent.
We describe the process of living particles by the measure-valued process , where Here denotes the Dirac measure on , where is the Borelian -algebra in .
Notice that for , represents the number of living particles in the region at time .
The process is a Markov process called Branching Diffusion Process. For existence and properties see, for example, . This process takes values in a closed subspace of , the space of finite Borel positive measures on .
Denote by the set of bounded and continuous functions on . For every , we define the norm .
For every and , measurable we set We will note for and the covariance process and the quadratic covariance process. Also is the projection of in the sense described in .
We introduce the following spaces: : class of right continuous-adapted processes with left limits and with finite variations on finite intervals starting at the origin at time 0; : class of processes in with nondecreasing trajectories; : class of processes in with , where Var() is the variation process associated to ; : class of processes in with ; : class of uniformly integrable martingales.
Also , , , , and are the corresponding local classes.
We take as the canonical process in the stochastic basis , where is a given initial configuration, following its usual construction.
By assuming that the functions driving the model depend on an unknown parameter , a statistical model associate to the process is considered.
More specifically let be an open and convex set representing the parametric space and assume that , , and depend on a parameter , then we have Here is the space of real-valued matrices. When no confusion is possible we will note by a norm in the space as well as the Euclidean norm in . These functions define, for a given initial configuration and any parameter , a probability in the same way is constructed.
Suppose now these functions satisfy the following properties for every . Lipshitz Local Condition. For all , there exists a constant such that Linear Growth Condition. There exists a constant nondepending on such that is an invertible matrix for all , hence is symmetric and positive definite. For all we have Let then , , and belong to with There exist constants and such that
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 . In general, we cannot have absolute continuity if depends on the parameter . This seems to be a constrain of the likelihood approach but in some cases it is possible to estimate 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.
We assume that the whole process is observed on an interval ; 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 is the left limit of process at time .
Let’s denote by the times at which the jumps of the process take place, then, if at time a particle dies at position and has offsprings we have The space of jumps is a closed subset of defines as
Let also be the random measure associated with the jumps of given by Finally, for every optional function on and a random measure on we define the process by
3. Martingale Representation of the Process and Local Characteristics
We study now the local characteristics of the process through the real process .
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 in , the process is decomposed as
is a square integrable martingale with zero mean under and
is the infinitesimal generator of the common diffusion law followed by the particles and is optional on .
Here and represent the first derivative with respect to and the mixed second derivative with respect to and , respectively, whereas = .
By definition is a local martingale, where is the compensator of the process then by adding and subtracting we have the following.
Corollary 3.2. For every the process is, under , a square integrable local martingale with zero mean and quadratic characteristic: Here,
Proposition 3.3. Let be a real-adapted process, h a truncating function, continuous, continuous, and a random measure in such that . Let . Then is a semimartingale with local characteristics with respect to a truncating function if and only if for every the process is a local martingale.
Proof. It is enough to see that is a continuous process and
Also, where is a nonnegative process then . Moreover, it is continuous therefore predictable and it belongs to .
We have the following result.
Theorem 3.4. For any and there exist a probability on as stochastic basis such that . We have a.s. and which are the local characteristics of with respect to for any . The restriction to is the only probability in the filtered space with these local characteristics. Here are given, for any truncating function by and on as or equivalently, for every optional function on :
Proof. From [5, Theorem 3.1], or [6, Chapter 5], we have the existence of a probability measure in making (2.5) a BDP with infinitesimal generator where
Moreover, for every non negative function and we have that
is a local martingale with respect to .
We can write (3.17) as 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 process based on a Girsanov theorem for semi-martingales.
As reference measure we take the one determined by that is, particles moving according to independent Brownian motions without drifts. In the sequel, as we start from a fix deterministic configuration , we will drop the dependence on , then we denote by and the respective probabilities generated by the reference measure and the functions given in (2.8) according to Theorem 3.4. We will denote by and the expectations under and , 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 , absolutely continuous with respect to , with the same local characteristics than , therefore and are a.s. equal.
Next, we define the function as whenever and zero are otherwise.
Also we define the following processes on : Note that is well defined on the basis . Indeed, then and it is predictable.
Moreover, Then is a purely discontinuous local martingale on .
The first term in is a local continuous martingale so the process is a local martingale. Their jumps have the form then
Hence, , the Doleans-Dole exponential local martingale of , is a local martingale on the same basis. Also -. and .
Let be now a sequence of local stopping times for ; we note by the restriction of to the and we define on it the probability measure as , where is the process stopped at time .
We have the following result.
Proof. Let’s note by the local characteristics of under the measure .
First, note that if in is an optional process then hence and in a similar way we have .
On the other hand, then and Thus, we have that for every -measurable function , or equivalently Hence, According to [10, Theorem ], is a version of the conditional expectation and consequently is a version of the compensator of on the basis . Then we have .
Next, we define and , where is the process stopped at time .
We can see that . Indeed, and its jumps are bounded; hence combining Theorem , Lemma in  we have that Moreover, and then We can write
As we have
From (3.1), with We identify and 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 we have for any that with density given by (4.8).
The log-likelihood is given by where is the number of jumps before time , and is the jump corresponding to time .
Here means that is locally absolutely continuous with respect to .
Proof. From Theorem 3.4 we have the existence of the probability measure with local characteristics given by (3.13) and (3.15); by Proposition 4.1 and are equal on the -algebra , therefore with density . By local uniqueness the result can be extended to the -algebra .
From (4.8) we can write but Then Neglecting terms nondepending on 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 .
We denote by the score process, where the dot means the gradient with respect to the parameter . It is well known that is a zero mean martingale under . Its quadratic variation is the observed incremental information and the associate variance process is the expected incremental information. We denote by the Fisher observed information. Finally, the expected information is , see, for example, . Among these four quantities we have the following relation: We have the following result.
Proposition 4.3. If in addition to (A1)–(A6), we assume the following conditions: The function is bounded, that is, for every . There exist constants , , , , , , and , such that for every , all , all , and all , the following inequalities are satisfied: then , and are given by, where
5. A Branching Ornstein-Uhlenbeck Process
We consider a BDP where particles move according to an Ornstein-Uhlenbeck process on , then The death rate does not depend on the position; hence every particle has an exponential distributed lifetime independently of the trajectory.
Its reproduction law satisfies where refers to the probability that a particle has offsprings. Then the parameter is where .
So we write From (4.28) we get
Noting that where is the number of particles alive on the interval then where is the number of ancestors and We finally have From (5.8) we obtain the maximum likelihood estimators: Here is the number of splitting on resulting in offsprings.
Moreover, we have the following results: 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 For small we take: where . As initial parameters we take
Numerical results from simulated trajectories are shown in Table 1. The particle system is observed until the time of the 1000th reproduction.
This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada. Also, the authors would like to thank the support received from the Asociación Mexicana de Cultura
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