The increments of the Wiener process are independent of C; hence 
E [ec d’ w(G)] = E [d w(t) c1] = 0 
Moreover, from the properties of the Wiener process 
il 
E [d w(T) dw (o)] = 0 
if dt and do have no parts in common; otherwise 
ae 
E [d w(T) dw (t)] = Ro dt 
where Rk, is the covariance matrix of the Wiener process w. The final 
expression for P is then 
t 
P(e) = OC, O) F OI, OY | o(t, T) R(t) oi (t, t) dt 
0 
(4.10) 
A differential equation for P can be obtained from this expression 
for P simply by differentiating 
dp fa T di 
@. Ee a(t, 0)| r o"(t, 0) + o(t, 0) roe, 0) 
fe 
+ O(t, t) R(t) Ot, @) 2 il ee R(t) Oo (, 1) aL 
0 
t 
+ | G(r, 1) R(t) SO (t, 1) at 
0 
Sil 
