CHAPTER 33
EXTENSION OF THE ITÔ INTEGRAL
In this chapter, we define the Itô integral for stochastic processes in a larger space.
33.1 Basic Concepts and Facts
Definition 33.1 (Stochastic Process Space 
. Let 
be a filtration under consideration. The space 
ad(Ω, L2[a, b]) is defined to be the space of all stochastic processes f(t, ω), t 
[a, b], ω Ω, satisfying the following conditions:
(a) f(t) is adapted to the 
t.
t.(b) 
a.s.
a.s.Definition 33.2 (Extension of the Itô Integral). Let {
t : a ≤ t ≤ b} be a filtration. Let {Bt : a ≤ t ≤ b} be a Brownian motion satisfying the following conditions:
(a) For each t, Bt is -measurable.
(b) For any s ≤ t, the random variable Bt − Bs is independent of s.
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