Estimation in Interacting Diffusions : Continuous and Discrete Sampling

Consistency and asymptotic normality of the sieve estimator and an approximate maximum likelihood estimator of the drift coefficient of an interacting particles of diffusions are studied. For the sieve estimator, observations are taken on a fixed time interval [0,T] and asymptotics are studied as the number of interacting particles increases with the dimension of the sieve. For the approximate maximum likelihood estimator, discrete observations are taken in a time interval [0,T] and asymptotics are studied as the number of interacting particles increases with the number of observation time points.


Introduction
Finite dimensional parameter estimation in one-dimensional stochastic differential equations from continuous and discrete observations by maximum likelihood and Bayes methods are extensively studied in Bishwal [1].Interacting particle systems of diffusions are important for modeling many complex phenomena, see Dawson [2] and Ligget [3].Interacting particle systems are useful in constructing particle filter algorithms for finance and computation of credit portfolio losses, see Carmona et al. [4].Grenander [5] introduced the method of sieves for estimating infinite dimensional parameters.Sieve estimation for linear stochastic differential equations is studied in Nguyen and Pham [6].Statistics for interacting particle models has not received much attention.Maximum likelihood estimation in interacting particle system of stochastic differential equations was studied in Kasonga [7].In this paper we study nonparametric and parametric estimation in interacting particle system of stochastic differential equations by the method of sieves and the approximate maximum likelihood method respectively.
Consider the model of interacting particles of diffusions satisfying the Itô stochastic differential equations p are unknown functions to be estimated based on observation of the process X in the time interval   The functions , ; =1,2, ,  are assumed to be known such that the system has a unique solution.
Here are some special cases: 1) Linear case: 2) Simple Mean-Field Model: Here the subsystems are interacting and exchangeable described by the system of SDE's The term containing can be viewed as an interaction between the subsystems that creates a tendency for the subsystems to relax toward the center of gravity of the ensemble.Here The case corresponds to sampling independent replications of the same process given below.
We need the following assumption and results to prove the main results.
Assumptions (A1): Suppose that are measurable and adapted processes satisfying .
and .
  0 = 0 I In the exchangeable case (A1) follows from McKean-Vlasov Law of Large Numbers.In particular, (A1) will be satisfied when which corresponds to the independent replicated sampling on   0,T .We also need the following version of Rebolledo's Central Limit Theorem for Martingales: Let , n M n Z   be a sequence of local square integrable martingales with .Suppose the following condition holds: , where   c t is a continuous increasing function with . Then , a continuous Gaussian martingale with zero mean and covariance function denotes the jump of M at the point s.

Sieve Estimator
Let P  and 0 be the probability distributions of where and are vectors and the matrix with general elements The restricted maximum likelihood estimator (sieve estimator) of  is

Properties of the Sieve Estimator
In this section we obtain consistency and asymptotic normality of the sieve estimator.We focus on the interacting and exchangeable cases described by the system of SDE's Proof.The method of proof is similar to Nguyen and Pham [6] by using assumption (A1).We omit the details.

Theorem 3.2 (Asymptotic Normality)
Let be such that Then under (A1), we have Proof.The method of proof is similar to Nguyen and Pham [6] by using Rebolledo's CLT for martingales.We omit the details.

Approximate Maximum Likelihood Estimator
In practice, one can not observe the diffusion process in continuous time.In this section we study parameter estimation based on observations at discrete time points.Let P  and 0 be the probability distributions of when  is the true parameter and = 0  respectively.It is well known that P  is absolutely continuous with respect to . 0 is the unknown parameter.The functions , ; = 1, , j n l p     are assumed to be known such that there exists a unique solution   X t to the above SDE.Approximate maximum likelihood estimation for the one dimensional case has been extensively studied, see Bishwal [1].The approximate log-likelihood based on observations It is known that AMLE is consistent as and T   0 T N  and satisfies LAN (local asymptotic normality) and asymptotically efficient as and T   T N , see Bishwal [1].In this paper we assume is fixed, and .

 T
The Radon-Nikodym derivative (likelihood) is given by Our aim is to estimate the parameter  based on The approximate log-likelihood based on observations Here we have used the approximation of the stochastic integral and the ordinary integral as in Bishwal [1].
Equating the derivative of the log-likelihood function to zero provides the estimating equations and the approximate maximum likelihood estimator (AMLE)

Properties of the Approximate Maximum Likelihood Estimator
In this section we obtain the consistency and asymptotic normality of the approximate maximum likelihood estimator.
Theorem 5.1 (Consistency) Under (A1), we have Proof.The method of proof is similar to Kasonga [7] by using assumption (A1) with the aid of discrete approximations of the stochastic and ordinary integrals in Bishwal [1].We omit the details.
Theorem 5.2 (Asymptotic Normality) Under (A1), we have Proof.The method of proof is similar to Kasonga [7] by using Rebolledo's central limit theorem for martingales with the aid of discrete approximations of the stochastic and ordinary integrals in Bishwal [1].We omit the details.

AML Estimation in Mean-Field Model
Let us consider approximate maximum likelihood estimator (AMLE) for the simple mean-field model  corresponds to sampling independent replications of Ornstein-Uhlenbeck processes on   0,T .Our parameter here is The AMLE is with


, the AMLE is given by

Remark 1 :
One can look at this problem in a different way.If one observes the first Fourier modes in the expansion of the solution (the finite dimensional projection of the corresponding random field) of a parabolic stochastic partial differential equation (SPDE) and let the dimension of the projection increase n n