Quality Learning Support For All
First time here? Checkout the FAQs!
x

*Math Image Search only works best with SINGLE, zoomed in, well cropped images of math. No selfies and diagrams please :)

For Example

Math Image Search 1
Math Image Search 2

Join the MathsGee Club for exclusive quizzes, courses and interactive content without ads.



ASK - ANSWER - COMMENT - VOTE - CREATE


Sites: Global Q&A | Wits | MathsGee Club | Joburg Libraries | StartUps | Zimbabwe | OER

MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

0 like 0 dislike
23 views
What is the filtered probability space?
in Data Science & Statistics by Platinum (91,198 points) | 23 views

1 Answer

0 like 0 dislike
Best answer
A filtered probability space, or stochastic basis, \(\left(\Omega, \mathcal{F},\left(\mathcal{F}_{t}\right)_{t \in T}, \mathbb{P}\right)\) consists of a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) and a filtration \(\left(\mathcal{F}_{t}\right)_{t \in T}\) contained in \(\mathcal{F} .\) Here, \(T\) is the time index set, and is an ordered set usually a subset of the real numbers \(-\) such that \(\mathcal{F}_{s} \subseteq \mathcal{F}_{t}\) for all \(s<t\) in \(T\)

Filtered probability spaces form the setting for defining and studying stochastic processes. A process \(X_{t}\) with time index \(t\) ranging over \(T\) is said to be adapted if \(X_{t}\) is an \(\mathcal{F}_{t}\)-measurable random variable for every \(t\)

When the index set \(T\) is an interval of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met.
- The probability space \((\Omega, \mathcal{F}, \mathbb{P})\) is complete.
- The \(\sigma\)-algebras \(\mathcal{F}_{t}\) contain all the sets in \(\mathcal{F}\) of zero probability.
- The filtration \(\mathcal{F}_{t}\) is right-continuous. That is, for every non-maximal \(t \in T\), the \(\sigma\)-algebra \(\mathcal{F}_{t+} \equiv \bigcap_{s>t} \mathcal{F}_{s}\) is equal to \(\mathcal{F}_{t}\)

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to \(\mathcal{F}_{t}\), and by replacing \(\mathcal{F}_{t}\) by \(\mathcal{F}_{t+} .\) This will then satisfy the usual conditions. In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met. However, the process of completing the probability space depends on the specific probability measure \(\mathbb{P}\) and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the \(\sigma\)-algebras \(\mathcal{F}_{t}\) are universally complete, which is still strong enough to apply much of the 'heavy machinery' of stochastic processes, such as the Doob-Meyer decomposition, section theorems, etc.
by Platinum (91,198 points)

Related questions

0 like 0 dislike
0 answers
1 like 0 dislike
0 answers
0 like 0 dislike
0 answers
0 like 0 dislike
1 answer
1 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Mar 27 in Data Science & Statistics by Pieter Gold Status (13,283 points) | 93 views
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Oct 16 in Data Science & Statistics by MathsGee Platinum (91,198 points) | 4 views
0 like 0 dislike
1 answer
1 like 0 dislike
1 answer
1 like 0 dislike
0 answers
asked Oct 2, 2020 in Data Science & Statistics by MathsGee Platinum (91,198 points) | 46 views
0 like 0 dislike
0 answers
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Apr 4 in Mathematics by MathsGee Platinum (91,198 points) | 35 views
0 like 1 dislike
2 answers
asked Jun 6, 2020 in Mathematics by MathsGee Platinum (91,198 points) | 308 views
1 like 0 dislike
1 answer
0 like 0 dislike
0 answers
0 like 0 dislike
1 answer
0 like 1 dislike
2 answers
0 like 0 dislike
3 answers

Join the MathsGee Answers & Explanations community and get study support for success - MathsGee Answers & Explanations provides answers to subject-specific educational questions for improved outcomes.



On MathsGee Answers & Explanations, you can:


  1. Ask questions
  2. Answer questions
  3. Comment on Answers
  4. Vote on Questions and Answers
  5. Donate to your favourite users
  6. Create/Take Live Video Lessons

Posting on MathsGee Answers & Explanations


  1. Remember the human
  2. Behave like you would in real life
  3. Look for the original source of content
  4. Search for duplicates before posting
  5. Read the community's rules

Join the MathsGee Club for exclusive quizzes, courses and interactive content without ads.



Wits Maths Questions

What would the 9 steps be to consider when creating a digital marketing strategy
Posted on Friday October 01, 2021

Nine important decisions to consider when developing a digital marketing strategy

Click Here To Read More.
 

Is the IMM Study Group platform easy to use?
Posted on Friday October 01, 2021

Is the IMM Study Group platform easy to use?

A grain company called Splitting My Grain analyses and tabulates the number of sales it fulfilled in 2020 in the table below:
Posted on Thursday September 30, 2021

A grain company called Splitting My Grain analyses and tabulates the number of sales it fulfilled in 2020 in the table below:

\[
\begin{array}{l|l|l|l|}
\hline {\text { Product }} & {\text {...

Click Here To Read More.
 

An online book shop called Paper Cut collects its home delivery data for its Gauteng depot, which is located in Boksburg.
Posted on Thursday September 30, 2021

An online book shop called Paper Cut collects its home delivery data for its Gauteng depot, which is located in Boksburg.

\[
\begin{array}{lll|}
\hline \text { Destination address city } &...

Click Here To Read More.
 

Bongani owns and operates a fuel garage. He compiled the table below showing the number of litres of each type of fuel sold per month in 2020 .
Posted on Thursday September 30, 2021

Bongani owns and operates a fuel garage. He compiled the table below showing the number of litres of each type of fuel sold per month in 2020 .

\[
\begin{array}{llll|}
\hline...

Click Here To Read More.
 

MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

Join the MathsGee Club for exclusive quizzes, courses and interactive content without ads.



Management Leadership | MathsGee ZOOM | eBook

Join the MathsGee Club for exclusive quizzes, courses and interactive content without ads.