In this chapter, we define the Itô integral for stochastic processes in a larger space.

**Definition 33.1** (Stochastic Process Space . Let be a filtration under consideration. The space _{ad}(Ω, *L*^{2}[*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}.

(b) a.s.

**Definition 33.2** (Extension of the Itô Integral). Let {_{t} : *a* ≤ *t* ≤ *b*} be a filtration. Let {*B*_{t} : *a* ≤ *t* ≤ *b*} be a Brownian motion satisfying the following conditions:

(a) For each *t, B*_{t} is -measurable.

(b) For any *s* ≤ *t*, the random variable *B*_{t} − *B*_{s} is independent of _{s}.

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