Lévy–Itô decomposition

E1020435

result in stochastic process theory theorem in probability theory

The Lévy–Itô decomposition is a fundamental result in probability theory that expresses any Lévy process as the sum of a Brownian motion with drift and a jump process constructed from a Poisson random measure.

Try in SPARQL Jump to: Statements Referenced by

Statements (44)

Predicate	Object
instanceOf	result in stochastic process theory ⓘ theorem in probability theory ⓘ
appliesTo	Lévy process NERFINISHED ⓘ
assumes	càdlàg sample paths of the Lévy process ⓘ stationary independent increments of the process ⓘ
characterizes	structure of Lévy processes ⓘ
clarifies	role of jumps in Lévy processes ⓘ separation of continuous and jump parts of a Lévy process ⓘ
decomposesInto	Brownian (Gaussian) part ⓘ drift part ⓘ large jumps part ⓘ small jumps part ⓘ
ensures	uniqueness of decomposition up to modification ⓘ
field	probability theory ⓘ stochastic analysis ⓘ stochastic processes ⓘ
generalizes	decomposition of Brownian motion into drift and martingale parts ⓘ
gives	canonical representation of Lévy processes ⓘ
implies	every Lévy process is sum of independent components ⓘ pathwise representation of Lévy processes ⓘ
involves	Brownian motion with drift ⓘ Gaussian component of a Lévy process ⓘ Lévy measure ⓘ Poisson random measure ⓘ compensated Poisson random measure ⓘ drift term ⓘ jump process ⓘ pure jump component of a Lévy process ⓘ
isFoundationFor	construction of stochastic integrals with respect to Lévy processes ⓘ theory of jump-diffusion SDEs ⓘ
isNamedAfter	Kiyosi Itô NERFINISHED ⓘ Paul Lévy NERFINISHED ⓘ
isRelatedTo	Lévy triplet NERFINISHED ⓘ Lévy–Khintchine formula NERFINISHED ⓘ
isTypicallyFormulatedOn	filtered probability space ⓘ
isUsedIn	construction of Lévy processes from Lévy triplets ⓘ infinite divisibility theory ⓘ jump-diffusion modeling ⓘ mathematical finance ⓘ
requires	compensator of the Poisson random measure ⓘ
statesThat	any Lévy process can be decomposed into a Brownian motion with drift plus a jump process ⓘ
uses	stochastic integral with respect to Brownian motion ⓘ stochastic integral with respect to Poisson random measure ⓘ
yields	martingale representation for compensated jump part ⓘ

Referenced by (1)

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

Paul Lévy → knownFor → Lévy–Itô decomposition ⓘ