for determining the best estimate is that the mean square estimation 
error be a minimum. This best estimate X(t) is dependent on the values 
of 2(@) yin gehe antervalyOR< sa < see ands elicansibeRprowed senatptyicma 
linear combination of the values of z on this interval. 
t 
R(t) = i Kes t) ol BC) (5.4) 
0 
Since z(t) is a stochastic variable, &(t) is a stochastic integral. 
Interpolation and extrapolation are two problems that are related 
to the filtering problem. The interpolation problem is one of estimating 
the state at some time T < t; the extrapolation problem is one of esti- 
mating it at some time tT > t. This latter problem is the one which is 
of interest to the stock market investor. 
The condition that X(t) is the best estimate from among all linear 
functionals of z(t) for the state vector x in the least squares sense is 
stated mathematically as follows. For every constant vector i and 
linear functional F, 
a D 
BL{ATGe(t) - (t))}7) < ELA (x(t) - F(2))}] — .5) 
where all variables have a zero mean. 
aISRCE) I) = BibaCe)] = BLE@) I = © 
Now set 
~ nw 
X=x- X 
where x is called the minimum error vector. 
we D7 <a Ee & @@) = SIA 
2 HO 37] & IA BA" GG) = DJ 
BO! @@) = D1 
54 
