Estimation in Financial Models - RiskLab

densities of Y and thus the log-likelihood function for based on observations
of Y 0 ;Y 1 ;:::;Y n , from which an estimation of can be obtained. As
Pedersen [59] points out, the unknown vector x 0 2 IR d should be treated as
a part of and is not to be chosen at random. The independence of X 0 and
e 0 implies Y 0 N k (T 0 x 0 + U 0 ;T 0 V 0 T T 0
+ W 0 ). Hence we can calculate x 0 by
x 0 = T ,1
0 (Y 0 ,U 0 ) a.s. if and only if k = d; V 0 = W 0 = 0 and T 0 is of full rank.
Furthermore, the starting points of the iterative procedure used to calculate
the densities p iji,1 depend on x 0 .
The Kalman-Bucy lter
Under some technical assumptions (see Pedersen [59], pp. 4{5), we have for
given observations y 0 ;y 1 ;:::;y n of Y 0 ;Y 1 ;:::;Y n
X i jY i = y i N d
i (y i ); i
; (3.29)
X i jY i,1 = y i,1 N d
Di i,1 (y i,1 )+S i ;R i
; (3.30)
Y i jY i,1 = y i,1 N d
Ti (D i i,1 (y i,1 )+S i )+U i ;T i R i T T
i + W i
; (3.31)
where R i = D i i,1 D T i
+ V i is positive denite, and where
0 (y 0 ) = x 0 + V 0 T T 0
T0 V 0 T T 0
+ W 0
,1
(y0 , T 0 x 0 , U 0 ); (3.32)
0 = V 0 , V 0 T T 0
i (y i ) = D i i,1 (y i,1 )+S i + R i T T
i
T0 V 0 T T ,1
0
+ W 0 T0 V 0 ; (3.33)
Ti R i Ti T ,1
+ W i
(y i , T i
Di i,1 (y i,1 )+S i
, Ui ) ; (3.34)
i = R i , R i T T
i (T i R i T T
i + W i ) ,1 T i R i : (3.35)
By means of the Kalman-Bucy lter we are now able to calculate l n () for
every xed 2 for given observations y 0 ;y 1 ;:::;y n of Y 0 ;Y 1 ;:::;Y n .
The iterative procedure (by means of the Kalman-Bucy lter)
(0) Calculate 0 (y 0 ) and 0 by means of formula (3.32) and (3.33).
(1) Given i,1 (y i,1 ) and i,1 the conditional distribution of Y i given Y i,1
is known, and so we can calculate the transition density p iji,1 (y i jy i,1 ; ).
(2) Calculate i (y i ) and i by means of formula (3.34) and (3.35) and
return to (1).
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