martingale representation theorem
E59640
The martingale representation theorem is a fundamental result in stochastic calculus stating that, under suitable conditions, every martingale can be expressed as a stochastic integral with respect to a Brownian motion (or more generally, a fundamental martingale).
All labels observed (2)
| Label | Occurrences |
|---|---|
| martingale representation theorem canonical | 3 |
| Martingales and stochastic integrals in the theory of continuous trading | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T478568 — 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: martingale representation theorem Context triple: [Girsanov theorem, relatedTo, martingale representation theorem]
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A.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
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B.
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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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.
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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E.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
- 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: martingale representation theorem Target entity description: The martingale representation theorem is a fundamental result in stochastic calculus stating that, under suitable conditions, every martingale can be expressed as a stochastic integral with respect to a Brownian motion (or more generally, a fundamental martingale).
-
A.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
-
B.
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.
-
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.
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.
-
E.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | theorem in stochastic calculus ⓘ |
| appliesTo |
continuous martingales
ⓘ
martingales adapted to the Brownian filtration ⓘ square-integrable martingales ⓘ |
| assumes |
completeness of probability space
ⓘ
probability space with filtration ⓘ right-continuous filtration ⓘ usual conditions on filtration ⓘ |
| conclusion |
Brownian motion
ⓘ
surface form:
Brownian motion is a fundamental martingale for its natural filtration
every square-integrable martingale can be represented as a stochastic integral ⓘ martingales are generated by a fundamental martingale ⓘ |
| dealsWith |
Brownian motion
ⓘ
adapted processes ⓘ filtrations ⓘ martingales ⓘ stochastic integrals ⓘ |
| field |
probability theory
ⓘ
stochastic analysis ⓘ stochastic calculus ⓘ |
| generalizationOf | representation of martingales in Brownian filtration ⓘ |
| hasVariant |
martingale representation for Lévy processes
ⓘ
martingale representation for Poisson random measures ⓘ martingale representation in general semimartingale setting ⓘ |
| implies |
any L2-martingale is an Itô integral with respect to Brownian motion
ⓘ
uniqueness of integrand up to indistinguishability ⓘ |
| importance |
central in theory of stochastic integration
ⓘ
fundamental structural result for martingales ⓘ key tool in continuous-time finance ⓘ |
| relatedTo |
Brownian filtration
ⓘ
Clark–Ocone formula ⓘ Doob–Meyer decomposition ⓘ
surface form:
Doob–Meyer decomposition theorem
Itô calculus ⓘ
surface form:
Itô integral
Itô’s lemma ⓘ
surface form:
Itô's lemma
|
| representationWithRespectTo |
Brownian motion
ⓘ
fundamental martingale ⓘ |
| requires |
existence of stochastic integral with respect to Brownian motion
ⓘ
square-integrability of the martingale ⓘ |
| typicalFormulation | every L2-martingale adapted to the Brownian filtration is an Itô integral of a predictable process ⓘ |
| usedIn |
Girsanov theorem applications
ⓘ
backward stochastic differential equations ⓘ completeness of financial markets ⓘ derivation of Black–Scholes formula ⓘ hedging theory ⓘ mathematical finance ⓘ stochastic control ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: martingale representation theorem Description of subject: The martingale representation theorem is a fundamental result in stochastic calculus stating that, under suitable conditions, every martingale can be expressed as a stochastic integral with respect to a Brownian motion (or more generally, a fundamental martingale).
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Martingales and stochastic integrals in the theory of continuous trading