Mathematical Statistics and Stochastic Processes

Chapter 13

Stochastic Integration and Diffusion Processes

13.1. Itô integral

Let (Wt, t ≥ 0) be a standard Wiener process with basis space , and be the class of random functions f such that:

1) , where λ is the Lebesgue measure on [a, b].

2) ∀t ∈ [a, b], f(t,·) is -measurable (i.e. f is nonanticipative with respect to W).

We propose to define an integral of the form:

We begin by defining the integral on the set of step functions belonging to :

where the intervals [ti,ti+1) are open on the right, except for the last, and fi is -measurable for all i.

We set:

LEMMA 13.1.–

1)

2)

PROOF.–

1)