For many stochastic differential equations often met in financial theory, it is the drift and the dispersion which are the principal parameters of the model. In such cases it is shown that the parameters can be estimated by ordinary methods from normal distribution theory.Stochastic differential equations; ML estimates; financial models

Erling B. Andersen

English

Research Papers in Economics

DISCUSSION PAPERS 
Department of Economics 
University of Copenhagen 
 
 
 
 
01-10 
 
 
Some Simple ML Estimators in Stochastic 
Differential Equations 
 
 
Erling B. Andersen 
 
 
 
 
 
Studiestræde 6, DK-1455 Copenhagen K., Denmark 
Tel. +45 35 32 30 82 - Fax +45 35 32 30 00 
http://www.econ.ku.dk 
 
 Some simple ML-estimators in stochastic
differential equations.
by Erling B. Andersen
University of Copenhagen
Abstract: For many stochastic differential equations, often met in financial theory, its is
the drift and the dispersion which are the principal parameters of the model. In such cases
it is shown that the parameters can be estimated by ordinary methods from normal dis-
tribution theory.
Key words: Stochastic differential equaltions. ML-estimates. Financial models.
1. Introduction
A general stochastic differential equation for a stochastic process W(t) is defined as
where f and g are known functions of their arguments and B(t) is a Brownian motion or a
dW(t) f t,W(t) dt g t,W(t) dB(t) ,
Wienerprocess.
There has been considerable interest over the last decade in estimation problems connected
with parametrized stochastic differential equations.
The basic approach has been to parametrize the stochastic process itself, i.e. to assume that
W(t) depends on a vector of parameters q =( q 1 , ... ,qk), such that the stochastic differential
equation becomes
(1) dW(t|q) f t,W(t|q)d t gt,W(t|q) dB(t) ,
For this parametric representation of a stochastic differential equation several estimation
procedures has been suggested. From an applied point of view, most of these are rather
complicated and involve advanced results from the theory of diffusion processes. Cf. e.g.
Kessler and Sørensen (1999) for important results and references.
In this paper, we consider a subclass of stochastic differential equations, which, although
simple in structure, covers several interesting cases from applications to financial theory.2
Consider thus the alternative formulations to (1)
(2) dW(t) af(t)dt sg(t)dB(t) ,
where f(t) and g(t) are known functions of t, or
(3) dW(t) af t,W(t) dt sg t,W(t) dB(t) ,
where again f and g are known functions of their arguments.
The purpose of this paper is to demonstrate, that for simple choices of the functions f and
g in (2) and (3), it is possible to derive simple estimators for a and s. In addition the
distributional properties of these estimators are easily derived from ordinary normal
distribution theory.
For simplicity, we shall mainly consider the equidistant case, where the process is
observed at timest=1 ,t=2 ,t=3a n ds oo n .T h eresults are easily extended to cases
with non-equidistant observation points in time. For the case in Section 2 it is demonstra-
ted, how this can be done.
We shall call the parameter a in (2) or (3) the drift and the parameter s in (2) or (3) the
dispersion of the process.
2. Constant drift and constant dispersion
If f(t) = 1 and g(t) = 1, we shall say that the process has constant drift and constant
dispersion. In this case Equation (2) takes the form
(4) dW(t) adt sdB(t) ,
or3
õ
ó
t
0
W(s)ds aõ
ó
t
0
ds sõ
ó
t
0
dB(t) .
The last integral is an Ito-integral, but in this case the function to be integrated is constant,
such that the solution is B(t) - B(0). Hence the solution to (4) is
since B(0) = 0, by the definition of a Brownian motion.
W(t) W(0) at s B(t) B(0) W(0) at sB(t),
Suppose now, that W(t) has been observed at timest=1 ,2 ,. . .,n ,a n dthat the observed
values are w(1), w(2), ... , w(n). With w(0) = 0, we can then form the new observations
y1, ... ,yn as
or
yi w(i) w(i 1) a(i (i 1)) s b(i) b(i 1) ,
(5) yi a s b(i) b(i 1) , i 1,...,n .
where b(i) is the realized value of the Brownian motion B(i) at time i.
The random variables Y1,...,Yn corresponding to the observed y’s thus have the common
distribution
(6) Yi ~ N a,s2 .
The variance of Yi follows from the fact, that for a Brownian motion the variance at time t
is var[B(t)]=t, and the covariance between time t and time s,t<s ,i s
Accordingly
cov B(t),B(s) t.4
In addition the Y’s are independent, since Yi = W(i) - W(i-1) = a + s( B(i) - B(i-1) ) and
var B(i) B(i 1) i (i 1) 2(i 1) i (i 1) 1.
a Brownian motion has independent increments.
If we write w(i) = wi, the transformation from (y1, ... ,yn)t o( w 1 , ... ,wn) is one-one and
the differential element is
Hence with the proper substitutions
dy1...dyn
dw1...dwn
1.
This means that the likelihood function for the y’s is equal to the likelihood function for
fY y1,...,yn fW w1,...,wn .
the w’s. The estimation of a and s and the distridutional properties of these parameters
can thus be derived from the distribution of Y1, ... ,Yn, which is a set of independent,
identically and normally distributed random variables.
The ML-estimator of a is thus
(7) ˆ a y 1
n
n
i 1
yi
w(n) w(0)
n
w(n)
n
,
since w(0) = 0. The ML-estimator of s2 is
(8) ˆ s2 1
n
n
i 1
yi y
2 1
n
n
i 1
æ
ç
è
ö
÷
ø
w(i) w(n)
n
2
,
where
The distribution of the estimators follows from (6). Thus
w(i) w(i) w(i 1) .5
and the distribution of ˆ s is described by
ˆ a~N
æ
ç
ç
è
ö
÷
÷
ø
a ,s 2
n
where s2 is the unbiased estimator for s2. Hence
nˆ s 2
s 2
n 1
n
s2 n
s 2
(n 1)s2
s2
,
For non-equidistant time points t1, ... ,tn, the observations are w(t1),w(t2),...,w(tn). With
nˆ s 2
s 2
~c 2 (n 1) .
w(t0) = 0, we again base the estimations on the differences
or
yi w(ti) w(ti 1) a(ti ti 1) s b(ti) b(ti 1),
(9) yi w(ti) a ti s b(ti),
where
From (9) follows that
w(ti) w(ti) w(ti 1), t i t i t i1, b(ti) b(ti) b(ti 1).
and
E Yi a ti
since
var Yi s2 ti ti 1 2ti 1 s2 ti ti 1 s2 ti ,
In addition the Y’s are independent since a Brownian motion has independent increments.
cov B(ti),B(ti 1 ti 1 .6
The log-likelihoodfunction is accordingly
The ML-estimates for a and s2 are easily derived from ln L as
lnL n
2
ln(2p) n
2
ln(s2) 1
2
n
i 1
ln ti 1
2
n
i 1
w(ti) a ti
2
s2 ti
(10) ˆ a
i
w(ti)
i
ti
w(tn) w0
tn 0
w(tn w0
tn
and
(11) ˆ s2 1
n
n
i 1
w(ti) ˆ a ti
2
ti
.
Since w(t0)=0
(12) ˆ a
w(tn)
tn
,
and inserting (12) in (11)
In order to derive the distribution of ˆ s we rewrite (11) as
ˆ s2 1
n
n
i 1
1
ti
æ
ç
ç
è
ö
÷
÷
ø
w(ti)
w(tn)
tn
ti
2
.
Here
ˆ s2 1
n
n
i 1
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
w(ti)
ti
ˆ a ti
2
.7
while
var
é
ê
ê
ê
ë
ù
ú
ú
ú
û
W(ti)
ti
1
ti
tis2 s2 ,
E
é
ê
ê
ê
ë
ù
ú
ú
ú
û
W(ti)
ti
1
ti
tia tia ,
This means that we can perform the classical splitting of sum of squares, necessary to
obtain the distribution of the unbaised estimate for s2. Her we get
By the addition theorem for the c2-distribution
Q
n
i 1
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
w(ti)
ti
a ti
2
n
i 1
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
w(ti)
ti
ˆ a ti ˆ a ti a ti
2
n
i 1
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
w(ti)
ti
ˆ a ti
2
(ˆ a a) 2t n Q 1 Q 2 .
while obviously
Q
s2
~c 2 (n) ,
Hence by Cochrans theorem
Q2
s2
~c 2 (1) .8
or
Q1
s2
1
s2
n
i 1
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
w(ti)
ti
ˆ a ti
2
~c 2 (n 1).
The results for non-equidistant observed time points are thus similar to those for equi-
nˆ s 2
s 2
~c 2 (n 1) .
distantly observed time points.
3. The drift a function of t and constant dispersion
For these cases, we assume, that the drift is an integrable function f(t) of t, and that the
dispersion is constant, i.e. the stochastic differential equation is given by
By integration we then get
dW(t) af(t)dt sdB(t) .
i.e. with F(t) being the integral
W(t) W(0) aõ
ó
t
0
f(s)ds s B(t) B(0)
and since B(0) = 0
F(t) õ
ó
t
0
f(s)ds ,
W(t) W(0) aF(t) sB(t) .
Taking differences, we get witht=i=1 ,2 ,3 ,. . .9
where as before DF(i) = F(i) - F(i-1). The Y’s are independent, since the B-differences are
Yi W(i) W(i 1) W(i) a F(i) s B(i) ,
independent. With the zi = DF(i) being known ’explanatory variables and
the model is thus equivalent to an origo regression model with explanatory variables
B(i) ~ N(0,1) ,
z1,. . .,z n , and the estimates and the distributional properties follow from ordinary
regression analysis theory. This means that the ML-estimators are given by
and
ˆ a
i
ziy(i)
i
z
2
i
,
For the special case f(t) = t, we get
ˆ s2 1
n
æ
ç
ç
ç
ç
ç
è
ö
÷
÷
÷
÷
÷
ø
i
y
2
i
i
ziyi
i
z
2
i
.
or the midpoint of the interval from t-1 to t.
F(t) õ
ó
t
t 1
sds t2
2
(t 1)2
2
t 1
2
,
4. The drift and the dispersion proportional to t
In this case, we consider the stochastic differential equation
Here it pays to study directly the differences
dX(t) µtdt stdZ(t) .10
From known properties of Brownian motions, cf. Øksendahl (1998), Section 3.2, follows
õ
ó
t
t 1
dW(t) W(t) W(t 1) W(t) a õ
ó
t
t 1
sds s õ
ó
t
t 1
sdB(t)
aæ
ç
è
ö
÷
ø
t 1
2
s õ
ó
t
t 1
sdB(t)
(13) E
é
ê
ê
ê
ë
ù
ú
ú
ú
û
õ
ó
t
t 1
sdZ(s) 0.
Hence by Iso’s isometry
Accordingly the independent observations yi = w(i) - w(i-1) have distributions
var
é
ê
ê
ê
ë
ù
ú
ú
ú
û
õ
ó
t
t 1
sdZ(s) E
é
ê
ê
ê
ê
ë
ù
ú
ú
ú
ú
û
æ
ç
ç
ç
è
ö
÷
÷
÷
ø
õ
ó
t
t 1
sdZ(s)
2
E
é
ê
ê
ê
ë
ù
ú
ú
ú
û
õ
ó
t
t 1
s2ds
õ
ó
t
t 1
s 2ds 1
3
t 3 (t 1)3 t(t 1) 1
3
.
with zi = i - 1/2 and ci = i(i-1) - 1/2.
Yi ~ N azi,cis2 ,
The model is thus equivalent to a weighted origo-regression model with weights 1/ci and
’explanatory variables’ zi. The ML-estimates for a and s2 are, therefore,
ˆ a
i
1
ci
yizi
i
1
ci
z
2
i11
and
ˆ s2 1
n
æ
ç
ç
ç
ç
ç
è
ö
÷
÷
÷
÷
÷
ø
i
1
ci
y
2
i
i
1
ci
ziyi
i
1
ci
z
2
i
5. The drift and the dispersion proportional to W(t)
Consider finally the model
(14) dW(t) aW(t)dt sW(t)dB(t) .
where both the drift and the dispersion are proportional to the process itself. This model
can also be written as
In order to evaluate the left hand side, we must apply Ito’s lemma to the function lnW(t),
dW(t)
W(t)
adt sdB(t) ,
cf. Øksendahl (1998), Section 5.1, Example 5.1.1. From Ito’s lemma, we get
(15) d lnW(t) dW(t)
W(t)
1
2
æ
ç
ç
è
ö
÷
÷
ø
1
W(t)2
dW(t)
2
But according to Ito’s identities and (14)
Thus from (15)
dW(t)
2 a2W(t)2(dt)2 s2W(t)2 dB(t)
2
2aW(t)sdtdB(t) s2W(t)2dt .
If we, therefore, use the observed values
d lnW(t) dW(t)
W(t)
1
2
s2dt æ
ç
è
ö
÷
ø
a 1
2
s2 dt sdB(t) .12
of the logarithmically transformered variables as observations, with the new trendparameter
yi lnw(i) lnw(i 1)
we get the ML-estimates
a1 a 1
2
s2 .
and
ˆ a1 y,
for a1, and for s2 .
ˆ s2 1
n
n
i
yi y
2
.
The estimate for the origional a becomes
ˆ a ˆ a1
1
2
ˆ s2 .
6. References
Kessler, M. and Sørensen, M. (1999): Estimating equations based on eigenfunctions for a
discretely observed diffusion process. Bernoulli. 5, 299-314.
Øksendahl, B. (1998): Stochastic Differential Equations. An Introduction with Applications.
Fifth Edition. Berlin: Springer Verlag.

Estimating equations based on eigenfunctions for a discretely observed diffusion process.

Stochastic Differential Equations. An Introduction with Applications. Fifth Edition.

Some Simple ML Estimators in Stochastic Differential Equations

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Some Simple ML Estimators in Stochastic Differential Equations

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