The Itô integral defines an integral of a stochastic process with respect to a standard Brownian motion. The Itô integral is more general than the Wiener integral introduced in the previous chapter. In this chapter, we shall introduce the concept of Itô integrals.

**Definition 32.1** (Stochastic Process Space *L*^{2}_{ad}([*a, b*] × Ω)). Let be a filtration under consideration. The space *L*^{2}_{ad}([*a, b*] × Ω) 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) ^{b}_{a} *E*(|*f*(*t*)|^{2})d*t* < ∞.

**Definition 32.2** (Itô Integral of Step Stochastic Processes). 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 _{t}-measurable.

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

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