Itô integral

E351145

construction in stochastic calculus mathematical concept stochastic integral

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.

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All labels observed (2)

Label	Occurrences
Itô integral canonical	6
vector-valued Itô integral	1

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 ⓘ Itô integral self-linksurface differs ⓘ surface form: vector-valued 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 ⓘ

Referenced by (7)

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

Kiyoshi Itô → knownFor → Itô integral ⓘ

Kiyoshi Itô → notableConcept → Itô integral ⓘ

Clark–Ocone formula → relatedTo → Itô integral ⓘ

Wiener measure → associatedWith → Itô integral ⓘ

Stratonovich integral → comparedTo → Itô integral ⓘ

Itô integral → hasAlternativeFormulation → Itô integral self-linksurface differs ⓘ

this entity surface form: vector-valued Itô integral

Itô isometry → appliesTo → Itô integral ⓘ