Itô’s lemma

E59984

mathematical theorem result in stochastic calculus

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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Observed surface forms (5)

Surface form	Occurrences
Itô formula	1
Itô's lemma	1
Itô’s lemma for jump processes	1
multidimensional Itô’s lemma	1
time-dependent Itô’s lemma	1

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 self-linksurface differs ⓘ surface form: Itô’s lemma for jump processes Itô’s lemma self-linksurface differs ⓘ surface form: multidimensional Itô’s lemma Itô’s lemma self-linksurface differs ⓘ surface form: time-dependent 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 model ⓘ surface form: Black–Scholes–Merton 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 calculus ⓘ surface form: Itô integral Stratonovich integral ⓘ
requires	once continuously differentiable functions in time ⓘ twice continuously differentiable functions in space ⓘ
usesConcept	adapted process ⓘ filtration ⓘ martingale ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Itô calculus → coreConcept → Itô’s lemma ⓘ

Itô processes → enables → Itô’s lemma ⓘ

subject surface form: Itô process

this entity surface form: Itô formula

Itô’s lemma → hasVariant → Itô’s lemma self-linksurface differs ⓘ

this entity surface form: multidimensional Itô’s lemma

Itô’s lemma → hasVariant → Itô’s lemma self-linksurface differs ⓘ

this entity surface form: time-dependent Itô’s lemma

Itô’s lemma → hasVariant → Itô’s lemma self-linksurface differs ⓘ

this entity surface form: Itô’s lemma for jump processes

Kiyoshi Itô → knownFor → Itô’s lemma ⓘ

Kiyoshi Itô → notableConcept → Itô’s lemma ⓘ

martingale representation theorem → relatedTo → Itô’s lemma ⓘ

subject surface form: Martingale representation theorem

this entity surface form: Itô's lemma