Applied Mathematics, 2011, 2, 1154-1158
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
E-mail: 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, Mean-Field Model, Large Interacting Systems, Diffusion
Process, Discrete Observations, Approximate Maximum Likelihood Estimation, Sieve Estimation
1. Introduction
Finite dimensional parameter estimation in one-dimen-
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
,
X
ttXtXtW t
jn
 
'
where n
 

12
=,,,
X
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
X
in the time interval
0,T. Let
=,,,ttt t

12
p

and

,
'
jp12
=,,
jj
j.
x
xx

x
The processes
,=1,2, ,
j
X
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 Mean-Field 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
X
ttXtttXt Xtt Wt
Xxj n


where
 
1
=1
=,
n
nj
j,
X
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 ,
X
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

'
I
t
is increasing for all
p
R
and .

0=0I
In the exchangeable case (A1) follows from McKean-
Vlasov Law of Large Numbers. In particular, (A1) will
be satisfied when

=
j
llj
X
X

and

=
j
j
X
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
M
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
M
, a continuous
Gaussian martingale with zero mean and covariance
function
 
,=,, 0,
K
st
=
c ststT where
s
ss
M
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
X
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

X
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
A
B
Since

n
A
is invertible almost surely,



1
ˆ=.
nnn
A
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
X
ttXtttXt Xt t
Wt
Xxjn


where
 
1
=1
=,
n
nj
j,
X
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
X
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
,
X
tXttXtW
jn
 
t
where
 

12
=,,,'
n
X
tXtXt Xt and
;0,= 1, 2,,t jn
j are independent Wiener
processes. Here
Wt
12
=,,,
p
 
,, =1,
jl j
is the unknown
parameter. The functions ,; =1,,jnl p

are assumed to be known such that there exists a unique
solution
X
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 log-likelihood based on observations
12 1
,,, ,=1,2,,
N11
X
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 Radon-Nikodym 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
X
tXtXt
 
 

 t
Our aim is to estimate the parameter
based on
particles
n

12
,,,n
XX X
 of

X
t on
0,T.
The approximate log-likelihood based on observations
12
,,, ,=1,2,
jj jN,
X
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 log-likelihood 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 Mean-Field Model
Let us consider approximate maximum likelihood esti-
mator (AMLE) for the simple mean-field model
 


 
d= ddd
0=0,=1,2,,
jjjn j
jj
,
X
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 Ornstein-Uhlenbeck 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
F
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


 
=
F
TGT
IT GT GT



with



2
2
0
=e1
2
T,
F
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 Ornstein-Uhlenbeck 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 Ornstein-Uhlenbeck 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, Springer-Verlag, Berlin Hiedelberg, 2008.
[2] D. Dawson, “Critical Dynamics and Fluctuations for a
Mean-Field Model of Cooperative Behavior,” Journal of
Statistical Physics, Vol. 31, No. 1, 1983, pp. 29-85.
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,” Springer-Ver-
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. 613-633. doi:10.1007/s00780-009-0098-8
=JT
F
TGTGT
, hence under 0
H
,


,ˆn has approximately

0,1N
nN TN
nJ
[5] U. Grenander, “Abstract Inference,” Wiley, New York,
1981.
distribution where
  



,,,
,
,
=nNnN nN
nN
nN
F
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. 603-611. doi:10.1137/0320045

J
T

of based on discrete observations with

P
,nN
J
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. 865-875.
doi:10.1137/0150050
where
is the chosen size of the test and is normal
quantile.
z