t 
B, | f CC) ad vo | = E[w(t)] - ®(t, 0) E[w(0)] 
0 
va 
0 
t 
=m(t) - O(t, 0) m(O) + { OES) ANG) eam! Ge) acic 
0 
Ol Can ye nA Ge) aa 
Hence 
t t 
E i OlG=aes) mec vo] = | OES eaccleanmn Gr) (4.6) 
0 
The properties of the solution of the stochastic differential 
equation (4.4) will now be investigated. Since x is a linear function 
of a normal process, it is also normal and can be characterized com- 
pletely by the mean value function and the covariance function. Since 
the expected value of the Wiener process w(t) is zero, 
t 
O(t, 0) Elc] + E | O(t, T) d w(t) 
0 
E[x(t) ] 
O(t, 0) My 
where My is the expected value of the initial condition c. Hence 
m, (t) = E[x(t)] = @(t, 0) My (4.7) 
Taking derivatives yields 
aoe = GE PE, 0) my = ACE) (te, 0) my = ACE) m, (4.8) 
49 
