By substituting (2.111) into (2.110) we have
where, in the second equality the variable changes s1 = s + u, s2 = u are made. Furthermore, by making the variable change into the inner integral in (3.33), we have
where is defined in (2.113).
By substituting (3.34) into (3.33), we obtain (2.112).
Assumption (1.47) allows to interchange the order of integral and expectation operators (Fubini and Tonelli Theorem (Champeney 1990, Chapter 3)).
From (2.112) with τ1 = τ and τ2 = 0 it follows that
from which (2.117) immediately follows.
To obtain (3.35), under assumptions (2.114)–(2.116), the order of
operations can be interchanged. In fact, the following inequalities hold
(3.36)
and
(3.37)
independent of T.
Therefore, conditions (2.114)–(2.116) allow to use the Fubini and Tonelli theorem (Champeney 1990, Chapter 3) to interchange the order of integrals in dt and ds for T finite. Moreover, the same conditions are sufficient to apply the dominated convergence theorem (Champeney 1990, Chapter 4) to interchange the order of limit and integral operations.
18.116.60.158