The integral 
t 
O®(t, T) d w(t) 
0 
is a stochastic integral. Since the transport matrix ®(t, T) is 
deterministic and has continuous derivatives, one way of defining 
this integral is through integration by parts. 
t 
| O(t, tT) d w(t) = O(t, t) w(t) - S(t, 0) wO) 
0 
t 
a® 
- J el Ce) awiGe) adar 
It follows from (1.15) and other properties of the transport matrix 
that 
t 
E 
| Oe, wt) a wie) = we) = ©Ces ©) w(O) a | O(t, tT) A(t) w(t) dt 
0 0 
(4.5) 
The integral on the right exist for almost all sample functions since 
the sample functions of w(t) are almost all continuous. This way of 
defining the integral has the desirable feature that the integral can 
be interpreted as an integral of sample functions. It does not, how- 
ever, preserve the intuitive idea that the integral is a limit of sums 
of independent random variables nor can it be extended to the case 
where ® is stochastic. Doob gives a more formal definition of the 
integral together with detailed proofs of its stochastic properties. 
The expected value of this integral is computed as follows: 
48 
