Clark–Ocone formula
E284688
The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Clark–Ocone formula canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2631546 — 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: Clark–Ocone formula Context triple: [Martingale representation theorem, relatedTo, Clark–Ocone formula]
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A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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B.
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.
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C.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
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D.
Dynkin formula
Dynkin formula is a fundamental result in the theory of Markov processes that expresses the expected value of a function of the process at a stopping time in terms of its generator and an integral over time.
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E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- 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: Clark–Ocone formula Target entity description: The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
B.
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.
-
C.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
-
D.
Dynkin formula
Dynkin formula is a fundamental result in the theory of Markov processes that expresses the expected value of a function of the process at a stopping time in terms of its generator and an integral over time.
-
E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in Malliavin calculus ⓘ result in stochastic calculus ⓘ |
| appliesTo |
functionals of Brownian motion
ⓘ
square-integrable random variables ⓘ |
| assumes |
adaptedness to Brownian filtration
ⓘ
square-integrability ⓘ |
| characterizes | square-integrable functionals as stochastic integrals plus constants ⓘ |
| context |
Brownian filtration
ⓘ
Wiener space ⓘ |
| field |
Malliavin calculus
ⓘ
probability theory ⓘ stochastic analysis ⓘ |
| formalSetting | L^2 space of the underlying probability space ⓘ |
| generalizationOf | martingale representation for Brownian motion ⓘ |
| gives |
integral representation of random variables
ⓘ
martingale representation ⓘ |
| hasComponent |
constant term equal to the expectation of the variable
ⓘ
stochastic integral term with predictable integrand ⓘ |
| hasVersion |
formula for Brownian motion in \/R
ⓘ
formula for Poisson random measures ⓘ formula for multidimensional Brownian motion ⓘ formula under change of measure ⓘ |
| integrandGivenBy | conditional expectation of the Malliavin derivative given the filtration ⓘ |
| involves | conditional expectation of the Malliavin derivative ⓘ |
| namedAfter |
Daniel Ocone
ⓘ
John Michael Clark ⓘ |
| relatedTo |
Girsanov theorem
ⓘ
Itô integral ⓘ Itô’s lemma ⓘ Malliavin calculus ⓘ
surface form:
Malliavin integration by parts
martingale representation theorem ⓘ |
| requires | Malliavin differentiability of the functional ⓘ |
| timePeriod | late 20th century ⓘ |
| type | representation theorem ⓘ |
| typicalAssumption | complete probability space with Brownian filtration ⓘ |
| usedFor |
computing Greeks via Malliavin calculus
ⓘ
explicit computation of hedging strategies ⓘ representation of payoffs in terms of Brownian motion ⓘ |
| usedIn |
derivative pricing
ⓘ
filtering theory ⓘ hedging theory ⓘ mathematical finance ⓘ sensitivity analysis of financial derivatives ⓘ stochastic control ⓘ |
| uses |
Brownian motion
ⓘ
Malliavin calculus ⓘ
surface form:
Malliavin derivative
stochastic integral ⓘ |
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: Clark–Ocone formula Description of subject: The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Martingale representation theorem