The condition that ¥ is the best estimate from among all linear func- 
tionals of z(t) leads to the result that E[x(t) aH ayy = 0. Hence 
P(e) EXC) S Ble) =2¢z)] wom) = R(t, &) Ra GE) 
From the stochastic integral, 
t 
(CE) = i K(t, r) dz(r) 
0 
iE 
d&(t) = K(t, t) dz(t) + | SRE) ae (r) at 
0 
(e 
=P HR) a(t) + | (A(t) K(t, x) - K(t, t) H(t) K(t, r)) dz(r) dt 
0 
or 
Ga(e) = Ate) SGe) de eo Ae Rot (d2(t) = Ge) Ae) ae) G13) 
Since z(t) and presumably dz(t) are known, this is a stochastic differ- 
ential equation for X(t). 
Note that 
dz(t) - H(t) &(t) dt 
dz(t) - H(t) x(t) dt + H(t) x(t) dt 
dv(t) + H(t) x(t) dt 
From this expression and (5.13), we get the following stochastic 
differential equation for x 
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