O'Reilly logo

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 33

EXTENSION OF THE ITÔ INTEGRAL

In this chapter, we define the Itô integral for stochastic processes in a larger space.

33.1 Basic Concepts and Facts

Definition 33.1 (Stochastic Process Space . Let be a filtration under consideration. The space ad(Ω, L2[a, b]) 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) a.s.

Definition 33.2 (Extension of the Itô Integral). Let {t : atb} be a filtration. Let {Bt : atb} be a Brownian motion satisfying the following conditions:

(a) For each t, Bt is -measurable.
(b) For any st, the random variable BtBs is independent of s.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required