Content deleted Content added
It is not clear how the second example is to be understood and a reference is not provided. |
m Task 18 (cosmetic): eval 2 templates: hyphenate params (4×); |
||
Line 4:
=== Discrete-time process ===
Given a [[filtered probability space]] <math>(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \in \mathbb{N}},\mathbb{P})</math>, then a stochastic process <math>(X_n)_{n \in \mathbb{N}}</math> is ''predictable'' if <math>X_{n+1}</math> is [[measurable function|measurable]] with respect to the [[sigma algebra|σ-algebra]] <math>\mathcal{F}_n</math> for each ''n''.<ref name="Zanten">{{cite web|title=An Introduction to Stochastic Processes in Continuous Time|first1=Harry|last1=van Zanten|date=November 8, 2004|url=http://www.cs.vu.nl/~rmeester/onderwijs/stochastic_processes/sp_new.pdf|format=pdf|
=== Continuous-time process ===
Given a filtered probability space <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})</math>, then a [[continuous-time stochastic process]] <math>(X_t)_{t \geq 0}</math> is ''predictable'' if <math>X</math>, considered as a mapping from <math>\Omega \times \mathbb{R}_{+} </math>, is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.<ref>{{cite web|title=Predictable processes: properties |url=http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf |format=pdf |
This [[σ-algebra]] is also called the '''predictable σ-algebra'''.
| |||