This yields the following differential equation for K(t, s): 
: ae S) = a(t) K(t, s) - K(t, t) W(t) K(t, s) (5-12) 
From the integral equation (5.8) for K(t, s) 
K(t, t) R(t) = Elx(t) x (t)] H(t) 
ie 
- I K(t, 1) H(t) Elx(x) x! (t)] wi (t) dr 
0 
On the other hand, 
t 
B[&(t) x! (t)] = E ! K(t, x) dz(r) e] 
0) 
t 
= | KG) HG) Bilas Ge) x! (t)] dr 
0) 
t 
+ | K(t, r) E[dv(r) eI 
0 
where the second integral vanishes; hence 
K(e, £) R(t) = Elx(e) x (e)] W(t) - ELR(t) x (t)] WC) 
El(x(t) - (t)) x! (t)] H(t) 
= {E[X(t) &1(t)] + ELR(t) #2 (t)]} w(t) 
59 
