Estimation in Financial Models - RiskLab

which in the case (3.6) has the form
@
@ (ln L T ()) =
Z T
0
b(X s )dX s ,
Solving @ @ (ln L T ()) = 0, we obtain the MLE
Z T
0
b 2 (X s )ds:
^ T =
R T
R
0
T
0
b(X s )dX s
b 2 (X s )ds
and hence, using equality (3.4), we have
^ T = 0 +
where 0 denotes the true value of the parameter.
Now we derive some properties of the MLE ^ T .
R T
0
b(X s )dW R
s
T
0
b 2 (X s )ds ; (3.7)
First, we treat the problem of the bias of an estimator. The bias in ^ T as an
estimator of 0 is dened as
b^ T
( 0 )=E 0 (^ T , 0 );
where E 0
denotes the expectation with respect to P 0 . By (3.7) we have
"R T
#
0
b(X s )dW s
E 0
^T = 0 + E 0 : (3.8)
b 2 (X s )ds
R T
0
Note that in the case of a constant drift coecient b(X t ) b
E
bW T
b 2 T = 1
bT E W T =0;
thus E 0
^T = 0 , so that ^ T is unbiased. In all other cases the estimator is
biased and in the case (3.6) we have under some regularity conditions the
equality
b^ T
( 0 )=
d Z T
,1
E 0 b 2 (X t )dt!
;
d 0 0
(see [51], x17.2 and x17.3, Theorem 17.2 and 17.3).
Now we turn to the properties of consistency and asymptotic normality.
Assuming some natural regularity conditions (see [38], p.83 and p.90, or [17],
p.500f, or [48], p.95), the stochastic dierential equation (3.4) has an ergodic
solution. For the ergodic theory we refer to e.g. [32], x18. Then the stationary
18