Applied Mathematics, 2011, 2, 11541158 doi:10.4236/am.2011.29160 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Estimation in Interacting Diffusions: Continuous and Discrete Sampling Jaya Prakash Narayan Bishwal Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, USA Email: J.Bishwal@uncc.edu Received July 14, 2011; revised August 3, 2011; accepted August 10, 2011 Abstract Consistency and asymptotic normality of the sieve estimator and an approximate maximum likelihood esti mator of the drift coefficient of an interacting particles of diffusions are studied. For the sieve estimator, ob servations 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, dis crete 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. Keywords: Stochastic Differential Equations, MeanField Model, Large Interacting Systems, Diffusion Process, Discrete Observations, Approximate Maximum Likelihood Estimation, Sieve Estimation 1. Introduction Finite dimensional parameter estimation in onedimen sional 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 esti mation for linear stochastic differential equations is studied in Nguyen and Pham [6]. Statistics for interacting particle models has not received much attention. Maxi mum likelihood estimation in interacting particle system of stochastic differential equations was studied in Ka songa [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 re spectively. Consider the model of interacting particles of diffusions satisfying the Itô stochastic differential equa tions n =1 d= d =1,2, , p jljlj j l , ttXtXtW t jn ' where n 12 =,,, tXtXt Xt and ;0,= 1, 2,,t jn j are independent Wiener processes. Here Wt 20,,d , lLTt = 1,,l p are unknown functions to be estimated based on observation of the process in the time interval 0,T. Let =,,,ttt t 12 p and , ' jp12 =,, jj j. xx x The processes ,=1,2, , j tj n are observed on 0,T. The functions ,; =1,2,, jj jn are assumed to be known such that the system has a unique solution. Here are some special cases: 1) Linear case: =1 d=dd,=1, 2,, p jl j l ttXttXtWt jn 2) Simple MeanField Model: Here the subsystems are interacting and exchangeable described by the system of SDE’s d= ddd 0=0,=1,2, , jj jn jj , j ttXtttXt Xtt Wt Xxj n where 1 =1 =, n nj j, tnXt tt and
J. P. N. BISHWAL1155 0t . 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 t =,tt t. The case corresponds to sampling independent replications of the same process given below. 0t 3) Independent case: d=d d,=1,2, jjjj , ttXttXtWtj n p We need the following assumption and results to prove the main results. Assumptions (A1): Suppose that are measurable and adapted processes satisfying 1 :=; =1,2,,; =1,2, jl jlj bssj nl 0 =1 1d.. nt jl jllm j bsbssctas asn n ,=1,2,lmp where lm ct are finite and con tinuous nonrandom functions of 0,tT. The limiting matrix is positive definite, =It ,=1,2,, lm lm p c t ' t is increasing for all R and . 0=0I In the exchangeable case (A1) follows from McKean Vlasov Law of Large Numbers. In particular, (A1) will be satisfied when = llj X and = j X 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 nZ be a sequence of local square integrable martingales with . Suppose the following condition holds: 0=0 n M 2>0 nn st EMsIMs for all 0,,> 0;tT and a.s. n Mt ct for all 0,tT, where ct is a continuous increasing function with . Then 0=0cD n M , a continuous Gaussian martingale with zero mean and covariance function ,=,, 0, st = c ststT where ss MM denotes the jump of M at the point s. 2. Sieve Estimator Let P and 0 be the probability distributions of P ,0, t tT when is the true parameter and =0 respectively. Since 20,,d ,LTt hence P is absolutely continuous with respect to . 0 P 2 0 =1 =1 0 2 0 =1=1=1 d=exp d d 1 2 d. pnT ljljj lj ppnT ljlj lmj jm m PtXt XtX P tXt Xt Xtt t Our aim is to estimate the function on 0,T based on interacting particles n 12 ,n XX X,, of t on 0,T. The log likelihood function is then 2 0 =1 =1 0 =1=1 =1 2 =d 1 2 d. pnT nljlj lj ppnT ljl lmj jjmm LtXtXtX t tXt XtXtt t () j Let be increasing sequence of subspaces of n V 20,LTt n V ,d with finite dimension such that 1 n is dense in dn 20,,dLTt. The method of sieve (see Grenander [5]) consists of maximizing n L on n. Let k V,k= 1, 2,, be a sequence of independent vectors of ,dt 20,LT such that 1,,dn form a basis of for all . Then for n V n k k d =1 n k ,= n V , we have d 2 , 0 =1 =1=1 d , 0 =1=1 =1=1 2 , =1 =d 1 2 d 1 =2 n pnn T lk kjljj lj k ppnn T lk kjl lmj k dn jjmmkk k ' nnn nn L tXtXtXt tXt XtXtt t BA where and , nn B n A are vectors and the matrix with general elements ,= 1, 2,, d kn k 2 0 =1 =1 =d pnT n kkjlj lj BtXtXt , j Xt t =1,2, ,dn k , 2 0 =1=1 =1 = d, n kr ppnT kjljjm r lmj A tXt XtXtt ,=1,2, ,d. n kr The restricted maximum likelihood estimator (sieve estimator) of is t d =1 ˆˆ = n nn kk k n where 12 ˆˆˆ ˆ =,,, n nnn dn Copyright © 2011 SciRes. AM
J. P. N. BISHWAL 1156 is the solution of ˆ= nn n B Since n is invertible almost surely, 1 ˆ=. nnn B 3. Properties of the Sieve Estimator In this section we obtain consistency and asymptotic nor mality of the sieve estimator. We focus on the interacting and exchangeable cases described by the system of SDE’s d=()d d, 0=0,=1,2, , jj jn j jj d ttXtttXt Xt t Wt Xxjn where 1 =1 =, n nj j, tnXt tt and . 0t Here . =,tt t 0 Theorem 3.1 (Consistency) Under (A1), we have 2 0 ˆd TnP ttt as and such that n dn 2 d0 n n. 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 12 d =,,n nnn n 22 n k d =1 n k n as . Then under (A1), we have d1 =1 ˆ0, nn D' n kk k k nNIT n as and d such that n 3 d0 n n. Proof. The method of proof is similar to Nguyen and Pham [6] by using Rebolledo’s CLT for martingales. We omit the details. 4. 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 P ,0, t tT when is the true parameter and =0 respectively. It is well known that P is absolutely continuous with respect to . 0 P The model is =1 d=d d =1,2, , p jljljj l , tXttXtW jn t where 12 =,,,' n tXtXt Xt and ;0,= 1, 2,,t jn j are independent Wiener processes. Here Wt 12 =,,, ,, =1, jl j is the unknown parameter. The functions ,; =1,,jnl p are assumed to be known such that there exists a unique solution t to the above SDE. Approximate maxi mum likelihood estimation for the one dimensional case =1n has been extensively studied, see Bishwal [1]. The approximate loglikelihood based on observations 12 1 ,,, ,=1,2,, N11 tXtXtjn with = i tiTN. It is known that AMLE is consistent as and T 0TN and satisfies LAN (local asy mptotic normality) and asymptotically efficient as and T 23 0 N n TN , see Bishwal [1]. In this paper we assume is fixed, and . T The RadonNikodym derivative (likelihood) is given by 2 0 =1 =1 0 2 0 =1 =1=1 d=exp d d 1d. 2 pnT ljl jj lj pp nT lm jljjm lm j PXtXtX t P tXtXt t Our aim is to estimate the parameter based on particles n 12 ,,,n XX X of t on 0,T. The approximate loglikelihood based on observations 12 ,,, ,=1,2, jj jN, tXtXt jn with = i tiTN is defined as , 2 11 =1=1 =1 2 11 =1=1=1 =1 11 = 1 2 . Nn pnN ljliji jiji lji pp nN lmjl iji lmji jmiii L XtXtXtX t Xt Xt Xttt 1 1 Here we have used the approximation of the stochastic integral and the ordinary integral as in Bishwal [1]. Equating the derivative of the loglikelihood function to zero provides the estimating equations 2 11 =1 =1 () 2 ,11 =1=1 =1 11 ˆ = , nN jlijij iji ji pnN n Nljl iji lji jmii i XtXtX tX t Xt Xt Xtt t =1,2, ,mp Copyright © 2011 SciRes. AM
J. P. N. BISHWAL1157 . and the approximate maximum likelihood estimator (AMLE) ,1 ,2, ˆˆˆ ˆ =,,, nnn n NNN Np 5. Properties of the Approximate Maximum Likelihood Estimator In this section we obtain the consistency and asymptotic normality of the approximate maximum likelihood esti mator. Theorem 5.1 (Consistency) Under (A1), we have ˆnP N as and . N n 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 1 ˆ0, nD N nNITN as and . n Proof. The method of proof is similar to Kasonga [7] by using Rebolledo’s central limit theorem for martin gales with the aid of discrete approximations of the sto chastic and ordinary integrals in Bishwal [1]. We omit the details. 6. AML Estimation in MeanField Model Let us consider approximate maximum likelihood esti mator (AMLE) for the simple meanfield model d= ddd 0=0,=1,2,, jjjn j jj , tXttXt Xt t Wt Xxj n where 1 =1 =,, n nj j Xt nXt and 0 . The case =0 corresponds to sampling independent repli cations of OrnsteinUhlenbeck processes on 0,T. Our parameter here is =, ˆ nn . The AMLE is with ˆ =, n NNN ˆ ,, ,, ˆ=, nnN nN N nN nN STRT TGT ,, ,, ,,, ˆ= nnN nNnNnN N nNnN nN GTSTFTRT FTGTGT where 1 ,1 =1 =1 = nN nNjiji ji ji STNXtXtXt , 1 11 =1 =1 =, nN nN jiniji ji ji RT N XtXtXtXt 1 1 2 1 ,1 =1 =1 =, nN nNj iii ji FT NXttt 2 1 ,11 =1 =1 =. nN nNj in iii ji GTNXt Xttt 1 Suppose 0 =1 10 n j jx n almost surely and 22 0 =1 10 n j jx n 2 0 almost surely as . Then the estimator n nP N as and n and N ITN 1 ˆn N nN 0, D as and n where = TGT IT GT GT with 2 2 0 =e1 2 T, TG T 2 2 20 2 e1 1 =. 22 4 T T eT GT In the classical case when =0 , the AMLE is given by 11 =1 =1 2 11 =1 =1 ˆ=. nN jiji ji ji n NnN jii i ji XtXt Xt Xtt t Sampling independent OrnsteinUhlenbeck pro cesses on n 0,T and letting and give weak consistency and asymptotic normality of the AMLE: n N nP ˆN and 22 0 2 ˆ0, e1 nD NT nN as and n. N 1 , 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 Copyright © 2011 SciRes. AM
J. P. N. BISHWAL Copyright © 2011 SciRes. AM 1158 7. References while remains fixed, the Fourier modes are indepen dent OrnsteinUhlenbeck processes, see Bishwal [1]. Another important point to be noted here is the connec tion between the method of sieves and the spectral Fourier asymptotics in SPDE. T [1] J. P. N. Bishwal, “Parameter Estimation in Stochastic Differential Equations,” Lecture Notes in Mathematics, Vol. 1923, SpringerVerlag, Berlin Hiedelberg, 2008. [2] D. Dawson, “Critical Dynamics and Fluctuations for a MeanField Model of Cooperative Behavior,” Journal of Statistical Physics, Vol. 31, No. 1, 1983, pp. 2985. doi:10.1007/BF01010922 Remark 2: Testing of hypothesis of noninteraction versus interaction of the subsystems, i.e., 0:=0H versus 1:H0 based on discrete observations of the system can be studied. Since ˆn nN has approximately distribution where 1 0,NJT [3] T. Ligget, “Interacting Particle Systems,” SpringerVer lag, New York, 1985. 1FT [4] R. Carmona, J. P. Fouque and D. Vestal, “Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses,” Finance and Stochastics, Vol. 13, No. 4, 2009, pp. 613633. doi:10.1007/s0078000900988 =JT TGTGT , hence under 0 , ,ˆn has approximately 0,1N nN TN nJ [5] U. Grenander, “Abstract Inference,” Wiley, New York, 1981. distribution where ,,, , , =nNnN nN nN nN TG TG T JT FT is an estimate [6] H. T. Nguyen and T. D. Pham, “Identification of Nonsta tionary Diffusion Model by the Method of Sieves,” SIAM Journal of Control and Optimization, Vol. 20, No. 5, 1982, pp. 603611. doi:10.1137/0320045 T of based on discrete observations with P ,nN TJT as and . Thus the null hypothesis is rejected if N n , ˆ> n 2Nz nN nJ T [7] R. A. Kasonga, “Maximum Likelihood Theory for Large Interacting Systems,” SIAM Journal on Applied Mathe matics, Vol. 50, No. 3, 1990, pp. 865875. doi:10.1137/0150050 where is the chosen size of the test and is normal quantile. z
