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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 13

Stochastic Integration and Diffusion Processes

13.1. Itô integral

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

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

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

We propose to define an integral of the form:

images

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

images

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

We set:

LEMMA 13.1.–

1)

2)

PROOF.–

1)

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