Itô integral
E351145
The Itô integral is a fundamental stochastic integral used in probability theory and mathematical finance to rigorously define integration with respect to Brownian motion and more general semimartingales.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Itô integral canonical | 7 |
| vector-valued Itô integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3365580 — 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 integral Context triple: [Kiyoshi Itô, knownFor, Itô integral]
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A.
Stratonovich integral
The Stratonovich integral is a formulation of stochastic integration that preserves the classical chain rule of calculus and is widely used in physics and engineering for modeling systems with noise.
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B.
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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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.
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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E.
Riemann–Stieltjes integral
The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
- 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 integral Target entity description: The Itô integral is a fundamental stochastic integral used in probability theory and mathematical finance to rigorously define integration with respect to Brownian motion and more general semimartingales.
-
A.
Stratonovich integral
The Stratonovich integral is a formulation of stochastic integration that preserves the classical chain rule of calculus and is widely used in physics and engineering for modeling systems with noise.
-
B.
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.
-
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.
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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E.
Riemann–Stieltjes integral
The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
construction in stochastic calculus
ⓘ
mathematical concept ⓘ stochastic integral ⓘ |
| appliesTo |
continuous local martingales
ⓘ
semimartingales ⓘ |
| basedOn | Brownian motion ⓘ |
| codomain | stochastic processes ⓘ |
| constructionMethod |
L2 limit of simple predictable integrals
ⓘ
approximation by step processes ⓘ |
| contrastedWith | Stratonovich integral ⓘ |
| definedOn | filtered probability space ⓘ |
| domain |
adapted stochastic processes
ⓘ
square-integrable predictable processes ⓘ |
| field |
mathematical finance
ⓘ
probability theory ⓘ stochastic analysis ⓘ |
| formalVariable |
integrand
ⓘ
integrator ⓘ |
| generalizationOf | Riemann–Stieltjes integral to stochastic processes ⓘ |
| hasAlternativeFormulation |
matrix-valued Itô integral
ⓘ
vector-valued Itô integral ⓘ
linked to:
Itô integral
|
| hasKeyResult |
Itô isometry
ⓘ
Itô’s lemma ⓘ martingale representation theorem ⓘ |
| hasProperty |
depends on filtration
ⓘ
integrator has unbounded variation almost surely ⓘ non-anticipative ⓘ |
| influenced |
modern quantitative finance
ⓘ
stochastic control theory ⓘ |
| introducedIn | 1940s ⓘ |
| namedAfter | Kiyoshi Itô ⓘ |
| relatedTo |
Doob–Meyer decomposition
ⓘ
local martingales ⓘ quadratic variation ⓘ |
| requires | filtration satisfying usual conditions ⓘ |
| satisfies |
isometry property
ⓘ
martingale property ⓘ |
| typicalIntegrator |
multi-dimensional Brownian motion
ⓘ
standard Wiener process ⓘ |
| usedFor |
defining martingale representations
ⓘ
defining stochastic differential equations ⓘ integration with respect to Brownian motion ⓘ integration with respect to semimartingales ⓘ modeling random processes in finance ⓘ pricing derivative securities ⓘ |
| usedIn |
Black–Scholes model
ⓘ
interest rate models ⓘ stochastic volatility models ⓘ |
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 integral Description of subject: The Itô integral is a fundamental stochastic integral used in probability theory and mathematical finance to rigorously define integration with respect to Brownian motion and more general semimartingales.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Itô integral