Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

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 L²_ad([a, b] × Ω)). Let be a filtration under consideration. The space L²_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)dt < ∞.

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 ...