where the last equality results from (5.7). Since this equation holds 
Eom allluronande ct sinyteheysimitcersvalen[|Onnatells 
i 
1 T , a T 
Kes Ss) R(s) = E[x(t) x (s)] H (s) - | K(t, r) H(t) E[x(r) x (s)] H (s) dr 
0 
(5.8) 
This is a nonhomogeneous integral equation for K(t, s). Its kernel 
is H(r) E[x(r) i) E-(@)- Since it corresponds to a positive definite 
quadratic form, all its eigenvalues are positive and the equation has a 
solution. Unfortunately, it is not possible to calculate K(t, s) from 
this equation because E[x(r) <(@)1)- the covariance of x(s), is unknown. 
A different equation for K(t, s) can be obtained from (5.8) by 
differentiating both sides of it with respect to t. 
oK(ES) p(s) = S- Elx(t) x"(s)] Hs) 
= K(e, t) H(t) Elx(t) x! (s)] H (s) 
t 
{ o Reet) HG) BERG) ©] &G) de 
0 t 
By (5.1) 
dx = Ax dt + dw 
Hence 
Diente) (ey) HUG) = AG) ie) 2G] ae ae 
+ E[dw(t) <i 
57 
