Estimation in Financial Models - RiskLab

The sequence hMi n is called the square characteristic or the quadratic variation
of M n . Here hMi n is
see [66], p.455.
hMi n =
nX
i=1
Hence, equation (3.65) can be written as
^ n , =
X 2 i,1;
M n
hMi n
: (3.66)
Recall the denition of the Fisher information in the continuous time case
(see 3.1.1, p.19). Here in discrete time the Fisher information equals
Direct calculation shows
hence
"
#
I n () =E , @2 ln p (x 1 ;:::;x n )
:
@ 2
I n () =E
X
X
2
i,1
;
I n () =E hMi n ;
and therefore hMi n is often called the stochastic Fisher information. Calculations
based on (3.63) show that for large n the Fisher information is
approximately
n
; jj < 1;
1, 2
Since
I n ()
8
><
>:
; jj =1;
;
( 2 ,1) 2 jj > 1:
n 2
2
2n
hMi n ,!1
P a:s:;
we can apply the `law of large numbers for square integrable martingales'
and obtain
M n
,! 0 P a:s:;
hMi n
see [66], p.487, Theorem 4. Thus, with (3.66) we conclude that the estimator
is strongly consistent, that is
as n tends to innity.
^ n ,!
P a.s.
As for asymptotic behaviour of the estimator we only state the results and
refer to [67] x5 for a detailed treatment.
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