Brownian filtration
E284689
Brownian filtration is the natural increasing family of σ-algebras generated by a Brownian motion, encoding all information revealed by the process up to each time.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Brownian filtration canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2631547 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Brownian filtration Context triple: [Martingale representation theorem, relatedTo, Brownian filtration]
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A.
Itô processes
Itô processes are a class of stochastic processes, typically modeled as solutions to stochastic differential equations, that form the fundamental objects of study in Itô calculus and modern stochastic analysis.
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B.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
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C.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
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D.
Brownian motion
Brownian motion is the random, jittery movement of microscopic particles suspended in a fluid, whose explanation provided key evidence for the existence of atoms and the molecular nature of matter.
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E.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Brownian filtration Target entity description: Brownian filtration is the natural increasing family of σ-algebras generated by a Brownian motion, encoding all information revealed by the process up to each time.
-
A.
Itô processes
Itô processes are a class of stochastic processes, typically modeled as solutions to stochastic differential equations, that form the fundamental objects of study in Itô calculus and modern stochastic analysis.
-
B.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
-
C.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
-
D.
Brownian motion
Brownian motion is the random, jittery movement of microscopic particles suspended in a fluid, whose explanation provided key evidence for the existence of atoms and the molecular nature of matter.
-
E.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
filtration
ⓘ
increasing family of sigma-algebras ⓘ stochastic process concept ⓘ |
| appearsIn |
Black–Scholes model
ⓘ
Kolmogorov extension framework for Brownian motion ⓘ construction of Itô integral ⓘ continuous-time martingale representation theorem ⓘ |
| associatedWith |
Brownian motion
ⓘ
Brownian motion ⓘ
surface form:
Wiener process
|
| definedOn | probability space ⓘ |
| domain | Wiener space ⓘ |
| encodes | information revealed by Brownian motion up to each time ⓘ |
| generatedBy |
coordinate maps of Brownian motion
ⓘ
sigma-algebras of Brownian motion up to time t ⓘ |
| hasProperty |
Brownian motion has continuous paths adapted to it
ⓘ
Brownian motion has independent increments relative to it ⓘ Brownian motion ⓘ
surface form:
Brownian motion has stationary increments relative to it
complete in the usual augmentation ⓘ right-continuous in the usual augmentation ⓘ |
| hasVersion |
completed Brownian filtration
ⓘ
usual augmentation of Brownian filtration ⓘ |
| is |
canonical filtration on Wiener space
ⓘ
natural filtration of a Brownian motion ⓘ smallest filtration making Brownian motion adapted ⓘ |
| isIncreasingIn | time ⓘ |
| makes |
Brownian motion a Markov process
ⓘ
Brownian motion a martingale ⓘ |
| relatedTo |
Chapman–Kolmogorov equation
ⓘ
surface form:
Markov property of Brownian motion
strong Markov property of Brownian motion ⓘ |
| satisfies |
F_s subset F_t for s ≤ t
ⓘ
usual conditions after augmentation ⓘ |
| timeIndexedBy | nonnegative real numbers ⓘ |
| usedIn |
Doob–Meyer decomposition
ⓘ
Girsanov theorem ⓘ Itô calculus ⓘ filtering theory ⓘ martingale theory ⓘ mathematical finance ⓘ optimal stopping problems ⓘ option pricing theory ⓘ representation of martingales ⓘ stochastic calculus ⓘ stochastic differential equations ⓘ |
| usedToDefine |
local martingales driven by Brownian motion
ⓘ
predictable processes with respect to Brownian motion ⓘ progressively measurable processes with respect to Brownian motion ⓘ stopping times for Brownian motion ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Brownian filtration Description of subject: Brownian filtration is the natural increasing family of σ-algebras generated by a Brownian motion, encoding all information revealed by the process up to each time.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Martingale representation theorem