Itô’s integral

Abstract

We are given a real Brownian motion B(t), t ≥ 0, on a probability space (Ω, ℱ, ℙ) and we denote by (ℱ _t ) its natural filtration. For any T > 0 we are going to define a stochastic integral

$$\int_0^T {F\left( s \right)d\,B\left( s \right)} ,$$

called Itô’s integral, where F is generally not deterministic as in the Wiener integral considered before, but instead it is a stochastic process fulfilling suitable conditions, roughly speaking such that F(t) is ℱ _t -measurable for all t ∈ [0, T] (we say that F is adapted to the filtration (ℱ _t )).