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 integration by parts ⓘ
linked to:
Malliavin calculus
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 derivative ⓘ
linked to:
Malliavin calculus
stochastic integral ⓘ |
How these facts were elicited
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Instruction
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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 linked to:
martingale representation theorem