Itô processes

E60316

mathematical object stochastic process

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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Observed surface forms (3)

Surface form	Occurrences
Itô process	1
Itô diffusion	1
On stochastic processes (seminal papers on stochastic calculus)	1

Statements (48)

Predicate	Object
instanceOf	mathematical object ⓘ stochastic process ⓘ
definedOn	filtered probability space ⓘ probability space ⓘ
enables	Itô’s lemma ⓘ surface form: Itô formula stochastic integration ⓘ
field	probability theory ⓘ stochastic analysis ⓘ stochastic calculus ⓘ
generalForm	X_t = X_0 + ∫_0^t a_s ds + ∫_0^t b_s dW_s ⓘ
hasCoefficient	diffusion coefficient ⓘ drift coefficient ⓘ
hasComponent	diffusion term ⓘ drift term ⓘ finite variation part ⓘ local martingale part ⓘ
hasDrivingProcess	Brownian motion ⓘ Brownian motion ⓘ surface form: Wiener process
hasMathematicalStructure	quadratic variation ⓘ
hasOperation	stochastic integral with respect to Brownian motion ⓘ
hasProperty	adapted to filtration ⓘ almost surely continuous paths ⓘ finite quadratic variation ⓘ semimartingale ⓘ
hasRepresentation	sum of local martingale and finite variation process ⓘ
namedAfter	Kiyoshi Itô ⓘ
relatedTo	Ornstein–Uhlenbeck process ⓘ Stratonovich process ⓘ geometric Brownian motion ⓘ local martingale ⓘ martingale ⓘ
satisfies	stochastic differential equation ⓘ
specialCase	Brownian motion ⓘ martingale with zero drift ⓘ
subclassOf	Markov process ⓘ continuous semimartingale ⓘ semimartingale ⓘ
usedIn	Itô calculus ⓘ filtering theory ⓘ mathematical finance ⓘ population dynamics ⓘ quantitative finance ⓘ statistical physics ⓘ stochastic control ⓘ
usedToModel	asset prices ⓘ diffusion phenomena ⓘ interest rates ⓘ volatility ⓘ

Referenced by (5)

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

Itô calculus → appliesTo → Itô processes ⓘ

Itô’s lemma → appliesTo → Itô processes ⓘ

Kolmogorov backward equation → appliesTo → Itô processes ⓘ

this entity surface form: Itô diffusion

Kiyoshi Itô → knownFor → Itô processes ⓘ

this entity surface form: Itô process

Kiyoshi Itô → notableWork → Itô processes ⓘ

this entity surface form: On stochastic processes (seminal papers on stochastic calculus)