# CHAPTER 32

# THE ITÔ INTEGRAL

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.

# 32.1 Basic Concepts and Facts

**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:

*f*(

*t*) is adapted to the

_{t}.

^{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:

*t*,

*B*

_{t}is

_{t}-measurable.

*s*≤

*t*, the random variable

*B*

_{t}−

*B*

_{s}is independent ...

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