Stochastic Filtering Theory pp 48-76 | Cite as

# Stochastic Integrals

Chapter

## Abstract

Let L denote the family of all real-valued functions

*Y*_{ t }(*ω*) defined on**R**_{+}×*Ω*which are measurable with respect to ℬ(**R**_{+}) × A and have the following properties:- 1.
*Y*= (*Y*_{ t }) is adapted to (G_{ t }). - 2.
For each

*ω*,the function*t*→*Y*_{ t }(*ω*) is left-continuous.

## Keywords

Simple Process Finite Interval Quadratic Variation Continuous Version Stochastic Integral
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## Notes

- Section 3.1 is based on the ideas of Dellacherie [7] and Courrège [5]. The proofs of the theorems stated in 3.1 are to be found in [7]. The process (
*(4)*of Theorem 3.1.4 is called the dual predictable projection of (U_{t}) by Dellacherie [7]. Section 3.2 is based on Meyer [43, 44] and Courrège [5]. A full discussion of Ito’s stochastic integral is given in Ito [20]. Lemma 3.3.1 is from Gihman and Skorohod [15]. Lemma 3.3.3 is given in Friedman [13].Google Scholar

## Copyright information

© Springer Science+Business Media New York 1980