Itô’s lemma
E59984
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Itô’s lemma canonical | 7 |
| Itô formula | 2 |
| Itô's lemma | 1 |
| Itô’s lemma for jump processes | 1 |
| multidimensional Itô’s lemma | 1 |
| time-dependent Itô’s lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T478444 — 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’s lemma Context triple: [Itô calculus, coreConcept, Itô’s lemma]
-
A.
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.
-
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.
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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D.
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.
-
E.
Black–Scholes model
The Black–Scholes model is a fundamental mathematical framework in financial economics for pricing options and other derivatives by modeling asset prices as stochastic processes.
- 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’s lemma Target entity description: Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
-
A.
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.
-
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.
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.
-
D.
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.
-
E.
Black–Scholes model
The Black–Scholes model is a fundamental mathematical framework in financial economics for pricing options and other derivatives by modeling asset prices as stochastic processes.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in stochastic calculus ⓘ |
| appliesTo |
Brownian motion
ⓘ
Itô processes ⓘ functions of stochastic processes ⓘ |
| assumes | semimartingale framework for general versions ⓘ |
| contrastsWith | ordinary chain rule without quadratic variation term ⓘ |
| coreIdea | function of a stochastic process has extra term from quadratic variation ⓘ |
| describes | stochastic chain rule ⓘ |
| field |
mathematical finance
ⓘ
probability theory ⓘ stochastic calculus ⓘ |
| generalizes | classical chain rule ⓘ |
| hasVariant |
Itô’s lemma for jump processes
ⓘ
linked to:
Itô’s lemma
multidimensional Itô’s lemma ⓘ
linked to:
Itô’s lemma
time-dependent Itô’s lemma ⓘ
linked to:
Itô’s lemma
|
| historicalPeriod | 20th century mathematics ⓘ |
| holdsAlmostSurely | with respect to underlying probability measure ⓘ |
| influenced | development of modern mathematical finance ⓘ |
| involves |
diffusion term
ⓘ
drift term ⓘ quadratic variation ⓘ second derivative with respect to state variable ⓘ |
| isFormulatedIn | continuous time ⓘ |
| isTaughtIn |
graduate probability courses
ⓘ
quantitative finance programs ⓘ |
| isUsedFor |
change of variables for Itô processes
ⓘ
computing dynamics of functions of Markov processes ⓘ deriving stochastic differential equations ⓘ deriving the Black–Scholes equation ⓘ pricing derivatives in finance ⓘ transforming stochastic differential equations ⓘ |
| isUsedIn |
Black–Scholes–Merton model
ⓘ
linked to:
Black–Scholes model
continuous-time portfolio theory ⓘ filtering theory ⓘ interest rate models ⓘ stochastic control ⓘ stochastic volatility models ⓘ |
| mathematicalDomain |
analysis
ⓘ
measure-theoretic probability ⓘ |
| namedAfter | Kiyoshi Itô ⓘ |
| relatesTo |
Itô integral
ⓘ
linked to:
Itô calculus
Stratonovich integral ⓘ |
| requires |
once continuously differentiable functions in time
ⓘ
twice continuously differentiable functions in space ⓘ |
| usesConcept |
adapted process
ⓘ
filtration ⓘ martingale ⓘ |
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’s lemma Description of subject: Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
martingale representation theorem
linked to: Itô’s lemma
linked to: Itô’s lemma
linked to: Itô’s lemma
linked to: Itô’s lemma
linked to: Itô’s lemma