Itô–Taylor expansion
E645107
generalization of Taylor series
mathematical concept
stochastic expansion
tool in stochastic analysis
The Itô–Taylor expansion is a stochastic generalization of the Taylor series that expresses solutions of stochastic differential equations as series involving iterated Itô integrals, forming the basis for higher-order numerical schemes.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Itô–Taylor expansion canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7150465 — 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: Itô–Taylor expansion Context triple: [Milstein method, relatedConcept, Itô–Taylor expansion]
-
A.
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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B.
Clark–Ocone formula
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.
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C.
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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D.
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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E.
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.
- 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: Itô–Taylor expansion Target entity description: The Itô–Taylor expansion is a stochastic generalization of the Taylor series that expresses solutions of stochastic differential equations as series involving iterated Itô integrals, forming the basis for higher-order numerical schemes.
-
A.
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.
-
B.
Clark–Ocone formula
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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
generalization of Taylor series
ⓘ
mathematical concept ⓘ stochastic expansion ⓘ tool in stochastic analysis ⓘ |
| accuracyCharacterization |
strong order of convergence
ⓘ
weak order of convergence ⓘ |
| appliesTo |
Itô stochastic differential equations
ⓘ
stochastic differential equations ⓘ |
| assumes |
adaptedness of coefficients to the filtration
ⓘ
existence and uniqueness of SDE solution ⓘ |
| basedOn |
Itô calculus
NERFINISHED
ⓘ
Itô integral NERFINISHED ⓘ |
| component |
diffusion term expansion
ⓘ
drift term expansion ⓘ multi-index notation for iterated integrals ⓘ multiple stochastic integrals ⓘ |
| dependsOn |
moments of the driving Wiener process
ⓘ
regularity of drift and diffusion coefficients ⓘ |
| documentedIn | Numerical Solution of Stochastic Differential Equations (Kloeden and Platen) NERFINISHED ⓘ |
| enables |
high-order strong numerical methods for SDEs
ⓘ
high-order weak numerical methods for SDEs ⓘ systematic derivation of stochastic Runge–Kutta schemes ⓘ |
| field |
numerical analysis
ⓘ
stochastic calculus ⓘ stochastic differential equations ⓘ |
| generalizes | Taylor series NERFINISHED ⓘ |
| hasVariant |
strong Itô–Taylor expansion
NERFINISHED
ⓘ
truncated Itô–Taylor scheme ⓘ weak Itô–Taylor expansion NERFINISHED ⓘ |
| purpose |
derive higher-order numerical schemes for SDEs
ⓘ
express solutions of stochastic differential equations as series ⓘ obtain strong approximations of SDE solutions ⓘ obtain weak approximations of SDE solutions ⓘ |
| relatedTo |
Euler–Maruyama scheme
NERFINISHED
ⓘ
Itô’s lemma NERFINISHED ⓘ Kloeden–Platen methods NERFINISHED ⓘ Milstein scheme NERFINISHED ⓘ stochastic Runge–Kutta methods NERFINISHED ⓘ stochastic Taylor formula ⓘ |
| typicalReference |
Eckhard Platen
NERFINISHED
ⓘ
Peter E. Kloeden NERFINISHED ⓘ |
| usedIn |
computational finance
ⓘ
engineering models with noise ⓘ numerical simulation of stochastic differential equations ⓘ stochastic modeling in biology ⓘ stochastic modeling in physics ⓘ |
| uses | iterated Itô integrals ⓘ |
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: Itô–Taylor expansion Description of subject: The Itô–Taylor expansion is a stochastic generalization of the Taylor series that expresses solutions of stochastic differential equations as series involving iterated Itô integrals, forming the basis for higher-order numerical schemes.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.