Asymptotic Inference for a Linear Sto chastic
Dierential Equation with Time Delay
Alexander A. Gushchin
Steklov Mathematical Institute
Gubkina 8
117966 Moscow GSP-1, Russia
UweKuchler
Institut f ur Mathematik
Humb oldt-Universitat zu Berlin
Unter den Linden 6
D-10099 Berlin, Germany
May 21, 1997
Abstract
For the stochastic dierential equation
dX
(
t
)=
f
aX
(
t
)+
bX
(
t
;
1)
g
dt
+
dW
(
t
)
t
0
the local asymptotic properties of the likelihoo d function are studied. They very
depend on the true value of the parameter
#
=(
a b
)
1)
.Eleven dierentcases
are possible if
#
runs through
R
2
.Let
^
#
T
be the maximum likelihoo d estimator
of
#
based on (
X
(
t
)
t
T
). Applications to the asymptotic behaviour of
^
#
T
as
T
!1
are given.
Keywords:
LAMN LAN LAQ likelihoo d function limit theorems for mar-
tingales local asymptotic prop erties, maximum likelihood estimator stochastic
dierential equations time delay
1 Intro duction
Assume (
W
(
t
)
t
0) is a real-valued standard Wiener pro cess,
a
and
b
are real
numbers and (
X
(
t
)
t
;
1) is a solution of
dX
(
t
)=
aX
(
t
)
dt
+
bX
(
t
;
1)
dt
+
dW
(
t
)
t
0
(1.1)
This is a revised version of the Discussion Paper 29/96 (SFB 373) \Asymptotic prop erties of
maximum likelihoo d estimators for a class of linear sto chastic dierential equations with time delay".
The pap er was printed using funds made available by the Deutsche Forschungsgemeinschaft
1)
denotes the transpose of the corresp onding vector or matrix
1
with some xed initial condition
X
(
t
)=
X
0
(
t
),
t
2
;
1
0], where
X
0
(
)isacontinuous
stochastic process independentof
W
(
). The solution (
X
(
t
)
t
;
1) of (1.1) exists, is
pathwise uniquely determined and can be represented as
X
(
t
)=
x
0
(
t
)
X
0
(0) +
b
Z
0
;
1
x
0
(
t
;
s
;
1)
X
0
(
s
)
ds
+
Z
t
0
x
0
(
t
;
s
)
dW
(
s
)
t
0
:
(1.2)
Obviously, it has continuous paths for
t
0 with probability one. Here (
x
0
(
t
)
t
;
1)
denotes the so-called fundamental solution of the deterministic equation
_
x
(
t
)=
ax
(
t
)+
bx
(
t
;
1)
t>
0
x
(
t
)=1
t
=0
x
(
t
)=0
t
2
;
1
0)
:
(1.3)
Equation (1.1) is a very sp ecial case of linear sto chastic dierential equations of the
type
dX
(
t
)=
Z
0
;
1
X
(
t
+
s
)
a
(
ds
)
dt
+
dM
(
t
)
t
0
(1.4)
where
a
(
) is an arbitrary function of nite variation on
;
1
0] and (
M
(
t
)
t
0) is,
e.g., a semimartingale, see Mohammed and Scheutzow (1990).
Assume, the solution (
X
(
t
)
t
2
;
1
T
]) of (1.1) for some nite
T >
0 has b een
observed, the parameters (
a b
) are unknown and have to be estimated. Then we
have a parametric problem, which generalizes the statistical problem of estimating the
parameter in Langevin's equation
dX
(
t
)=
aX
(
t
)
dt
+
dW
(
t
)
t
0 (1.5)
(see, e.g., Basawa and Prakasa Rao 1980). Estimation problems for sto chastic dieren-
tial equations with time delayhave b een considered in few pap ers up to now, see Dietz
(1992) and K uchler and Kutoyants (1996) and the references therein. The model we
consider seems to be of interest by the following reasons. First, it is a relatively simple
example exhibiting a variety of qualitatively dierent local asymptotic prop erties for
dierentvalues of the parameter. Second, the mo del already shows some typical ef-
fects app earing in estimation problems for equations with time-delayed terms. Third,
in contrast to more general delay mo dels, we are able to compute explicitly the rates of
convergence and the limit distributions of estimators for every value of the parameter.
The solutions of (1.1) form an exponen
tial family of continuous sto chastic pro cesses in
the sense of K uchler and Srensen (1989). Thus the maximum likelihood estimator
^
#
T
of
#
=(
a b
)
can b e expressed explicitly by
^
#
T
=(
I
0
T
)
;
1
V
0
T
where
V
0
T
denotes the vector
V
0
T
=
Z
T
0
X
(
t
)
dX
(
t
)
Z
T
0
X
(
t
;
1)
dX
(
t
)
!
2
and
I
0
T
is the observed Fisher information matrix given by
I
0
T
=
0
B
B
@
Z
T
0
X
2
(
t
)
dt
Z
T
0
X
(
t
)
X
(
t
;
1)
dt
Z
T
0
X
(
t
)
X
(
t
;
1)
dt
Z
T
0
X
2
(
t
;
1)
dt
1
C
C
A
:
The estimator
^
#
T
is calculated from the log-likelihood function
log
dP
#
T
dP
(0
0)
T
(
X
)=
#
V
0
T
;
1
2
#
I
0
T
# #
2
R
2
(see e.g. Liptser and Shiryayev 1977). Here
P
(
ab
)
T
is the measure on
C
(
;
1
T
]) generated
by the solution (
X
(
t
)
t
2
;
1
T
]) of (1.1).
The main purp ose of this paper is to study local asymptotic properties of the family
(
P
#
T
#
2
R
2
) and then to draw conclusions for properties of the estimator
^
#
T
when
T
!1
.
Since the log-likelihoods are quadratic in
#
for each
T>
0, it is not surprising that the
family (
P
#
T
),
T>
0, is lo cally asymptotically quadratic (LAQ) at every
#
0
2
R
2
, see
Section 2 (for the notion of LAQ see Le Cam and Yang, 1990 or Jeganathan 1995).
Namely,choose
#
0
=(
a b
)
2
R
2
arbitrary but xed and introduce
#
=
#
0
+
'
T
,
where
=(
)
2
R
2
,
'
T
=
'
T
(
#
0
) is a normalizing regular 2
2 matrix with
'
T
!
0as
T
!1
.Thenwe get
log
dP
#
T
dP
#
0
T
(
X
)=
V
T
;
1
2
I
T
(1.6)
where
V
T
=
Z
T
0
X
(
t
)
dW
(
t
)
Z
T
0
X
(
t
;
1)
dW
(
t
)
!
'
T
(1.7)
and
I
T
=
'
T
I
0
T
'
T
:
(1.8)
In view of (1.6), to proveLAQat
#
0
one has to choose the matrices
'
T
(
#
0
)insucha
way that (a) the vectors (
V
T
I
T
) are bounded in probabilityas
T
!1
(b) if (
V
T
n
I
T
n
)
converges in distribution to a limit (
V
1
I
1
) for a subsequence
f
T
n
g!1
, then
E
exp(
V
1
;
1
2
I
1
)=1
for every
2
R
2
(c) if
I
T
n
converges in distribution to a limit
I
1
for a subsequence
f
T
n
g!1
, then
I
1
is almost surely p ositive denite. Recall also that the imp ortant
special cases of LAQ are the lo cal asymptotic mixed normality (LAMN) and the local
asymptotic normality (LAN). LAMN at
#
0
means that (
V
T
I
T
)converges in distribu-
tion to (
I
1
=
2
1
Z I
1
)as
T
!1
, where the matrix
I
1
is almost surely p ositive denite
and
Z
is a standard Gaussian vector indep endentof
I
1
. If, moreover,
I
1
is nonran-
dom, then wehaveLANat
#
0
.
Note that condition (c) is imp ortant since otherwise we are not in a position even to
3
establish asymptotic properties of
^
#
T
(cf. Dietz 1992). In general, (c) cannot be reached
with matrices
'
T
being diagonal. We construct
'
T
as the product of two quadratic
matrices
'
(1)
T
and
'
(2)
T
,
'
T
=
'
(1)
T
'
(2)
T
, where
'
(1)
T
converges to a nonsingular limit as
T
!1
(the dep endence on
T
cannotbeavoided in general) and
'
(2)
T
is diagonal with
elements tending to zero, in most case with dierent rates.
It is obvious from (1.7 ), (1.8) and (1.2) that the properties of the fundamental solu-
tion
x
0
(
t
)for
t
!1
very inuence the limit properties of (
V
T
I
T
). Recall that for
Langevin's equation (
b
= 0), it holds
x
0
(
t
)=
e
at
, the solution (
X
(
t
)
t
0) is the
Ornstein{Uhlenbeck pro cess and there are exactly three relevant cases in considering
local asymptotic properties (
a<
0,
a
=0,
a>
0). In our case the picture turns out to
be much more rich. To specify
'
T
and to study the limit b ehaviour of (
V
T
I
T
)wehave
to distinguish eleven dierent cases for
#
0
. These cases will b e introduced as follows.
The b ehaviour of
x
0
(
) is connected with the set of (complex) solutions of the char-
acteristic equation
;
a
;
be
;
=0
:
(1.9)
It is easy to see that the set of solutions of (1.9) is countable innite (if
b
6
=0)
and that for every
c
2
R
1
the set
c
:=
f
2
j
Re
c
g
is nite. In particular,
v
0
:= max
f
Re
j
2
g
<
1
. Dene
v
1
:=max
f
Re
j
2
Re
<v
0
g
(max
=
;1
). One veries easily that if
2
then
!
2
and no other
2
with
Re
=Re
exists. The equation (1.9) has at most two real solutions. If there exists
a real solution
v
then the real part of every nonreal solution is strictly less than
v
.
Consequently, the only possible real solutions are
v
0
(if there is exactly one) or
v
0
and
v
1
(if there are two).
Wehave
v
0
2
if and only if
b
v
(
a
):=
;
e
a
;
1
(1.10)
otherwise there exists a unique
0
in with Re
0
=
v
0
and
0
:=Im
0
>
0. Further-
more in this case it holds
0
<
. Moreover, a second real solution exists, i.e.
v
1
2
,
if and only if
v
(
a
)
<b<
0.
For every
in denote by
m
(
) the multiplicityof
as a solution of (1.8). Wehave
m
(
) = 1 for all
2
except
=
a
;
1, which b elongs to if and only if
b
=
v
(
a
)
and which has multiplicitytwo, if
b
=
v
(
a
).
Additional information about the solutions of the equation (1.9) can be found in Hayes
(1950).
The following lemma is crucial for this note. It is based on the inverse Laplace trans-
form and Cauchy's residual theorem and it can be found in a slightly other form in
Myschkis (1972), see also Hale and Verduyn Lunel (1993). The pro of will b e sketched
in the appendix.
Lemma 1.1
For al l
c<v
0
the fundamental solution
x
0
(
)
of
(1.3)
can berepresented
in the form
x
0
(
t
)=
0
(
t
)
e
v
0
t
+
X
k
2
c
Re
k
<v
0
c
k
e
k
t
+
o
(
e
t
)
for
t
!1
,
(1.11)
4
where
<c
and
c
k
aresomeconstants. Here
0
(
t
)
equals
0
(
t
)=
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
1
v
0
;
a
+1
if
v
0
2
m
(
v
0
)=1
2
t
+
2
3
if
v
0
2
m
(
v
0
)=2
A
0
cos(
0
t
)+
B
0
sin(
0
t
)
if
v
0
=
2
A
0
=
2(
v
0
;
a
+1)
(
v
0
;
a
+1)
2
+
2
0
B
0
=
2
0
(
v
0
;
a
+1)
2
+
2
0
:
Remarks:
1) Note that the three cases for
0
correspond to
b>v
(
a
),
b
=
v
(
a
) and
b<v
(
a
),
respectively.
2) Recall that for Langevin's equation, i.e.
b
=0, wehave
b>v
(
a
) for every
a
2
R
1
.
In this case it holds =
f
a
g
and therefore
v
0
=
a
and
x
0
(
t
)=
e
at
.
3) If
v
0
2
,
m
(
v
0
) = 1 (and
b
6
= 0 to avoid the case from the previous remark),
then for our purp oses it is necessary to separate a further term from the sum in
(1.11). Weget
x
0
(
t
)=
1
v
0
;
a
+1
e
v
0
t
+
1
(
t
)
e
v
1
t
+
o
(
e
t
) for
t
!1
, (1.12)
where
<v
1
,
1
(
t
)=
8
>
>
>
<
>
>
>
:
1
v
1
;
a
+1
if
v
1
2
A
1
cos(
1
t
)+
A
2
sin(
1
t
) if
v
1
=
2
A
1
=
2(
v
1
;
a
+1)
(
v
1
;
a
+1)
2
+
2
1
B
1
=
2
1
(
v
1
;
a
+1)
2
+
2
1
:
Here
1
denotes the uniquely determined positivenumber suchthat
1
=
v
1
+
i
1
2
. (We note that
1
2
(
2
) in this case.)
The proof follows the line of the pro of of Lemma 1.1 (see the App endix) in an obvious
way.
As it was mentioned ab ove, the limit prop erties of (
V
T
I
T
) are dierentineleven cases.
Table 1 represents these cases. The rst column describes these cases in terms of
v
0
and
v
1
, and the relations (1.11 ) and (1.12) make clear a connection between our
classication and asymptotic prop erties of
x
0
(
). The second column characterizes the
cases in terms of
a
and
b
. The third column is just a notation for these cases which
will be used in the rest of the pap er. The functions
u
(
a
),
a<
1, and
w
(
a
),
a
2
R
1
,
5