What is a Filtered Probability Space?

In summary, the encircled part in the provided Wikipedia article explains the definition of the smallest sub-sigma-algebra that contains the union of a collection of sub-sigma-algebras for any given set. The symbol used to represent this is a "join," which is part of lattice theory, and its dual symbol is a "meet."
  • #1
woundedtiger4
188
0
atkpqw.jpg
 
Physics news on Phys.org
  • #2
For a collection [itex]\{\mathcal F_s\}_{s\in S}[/itex] of sub-[itex]\sigma[/itex]-algebras of [itex]\mathcal F[/itex], the set [itex]\bigvee_{s\in S} \mathcal F_s[/itex] is defined to be the smallest sub-[itex]\sigma[/itex]-algebra [itex]\mathcal G\subseteq \mathcal F[/itex] such that [itex]\mathcal F_s\subseteq\mathcal G[/itex] for every [itex]s\in S[/itex].
 
  • #3
economicsnerd said:
For a collection [itex]\{\mathcal F_s\}_{s\in S}[/itex] of sub-[itex]\sigma[/itex]-algebras of [itex]\mathcal F[/itex], the set [itex]\bigvee_{s\in S} \mathcal F_s[/itex] is defined to be the smallest sub-[itex]\sigma[/itex]-algebra [itex]\mathcal G\subseteq \mathcal F[/itex] such that [itex]\mathcal F_s\subseteq\mathcal G[/itex] for every [itex]s\in S[/itex].

at http://en.wikipedia.org/wiki/Filtration_(mathematics)#Measure_theory

the encircle part says:

rmnzph.jpg


Similarly in the following picture the encircle part

2ag5bnl.jpg


does the symbole "
10fuquf.jpg
" represent Union? I understood the notation from your reply but what is the name of the symbol? is it logical disjunction (though logical disjunction doesn't make sense here) or is it universal quantifier (though logical quantifier is turned A) or is it just a capital V (the 22nd alphabet)?
 
  • #4
The symbol is called a "join". It is a symbol from lattice theory. Here, just means the smallest sigma-algebra that contains the union.

The dual symbol is ##\bigwedge## and is called a meet.
 
  • #5
Thanks
 

FAQ: What is a Filtered Probability Space?

What is a filtered probability space?

A filtered probability space is a mathematical model used to describe the probability of events occurring over time. It consists of a sample space, a set of events, and a probability measure. The sample space represents all possible outcomes of an experiment, the events are subsets of the sample space, and the probability measure assigns a numerical value to each event representing the likelihood of its occurrence.

How is a filtered probability space different from a regular probability space?

A regular probability space does not take into account the concept of time, whereas a filtered probability space considers events that occur over time. This means that the events in a filtered probability space are organized into a sequence or "filtration" that represents the flow of time. This allows for a more accurate representation of the probability of events occurring over time.

What is the significance of a filtration in a filtered probability space?

A filtration is a sequence of events in a filtered probability space that represents the flow of time. It allows for the consideration of past events and the prediction of future events, making it a crucial component in understanding the probability of events occurring over time in a filtered probability space.

How are conditional probabilities calculated in a filtered probability space?

In a filtered probability space, conditional probabilities are calculated using the concept of a conditional expectation. This involves taking into account the events that have already occurred, as represented by the filtration, and using this information to calculate the probability of future events occurring. This allows for a more accurate estimation of the probability of events over time.

What are some real-world applications of filtered probability spaces?

Filtered probability spaces have many practical applications, including in finance, economics, and engineering. They are used to model stock prices, interest rates, and other financial variables that change over time. They are also used in predicting the behavior of complex systems, such as weather patterns and traffic flow. Additionally, filtered probability spaces are used in machine learning and artificial intelligence to make predictions based on historical data.

Similar threads

Replies
6
Views
1K
Replies
5
Views
3K
Replies
1
Views
950
Replies
1
Views
954
Replies
2
Views
1K
Replies
4
Views
1K
Back
Top