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.

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