Itô–Taylor expansion

E645107

generalization of Taylor series mathematical concept stochastic expansion tool in stochastic analysis

The Itô–Taylor expansion is a stochastic generalization of the Taylor series that expresses solutions of stochastic differential equations as series involving iterated Itô integrals, forming the basis for higher-order numerical schemes.

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Statements (47)

Predicate	Object
instanceOf	generalization of Taylor series ⓘ mathematical concept ⓘ stochastic expansion ⓘ tool in stochastic analysis ⓘ
accuracyCharacterization	strong order of convergence ⓘ weak order of convergence ⓘ
appliesTo	Itô stochastic differential equations ⓘ stochastic differential equations ⓘ
assumes	adaptedness of coefficients to the filtration ⓘ existence and uniqueness of SDE solution ⓘ
basedOn	Itô calculus NERFINISHED ⓘ Itô integral NERFINISHED ⓘ
component	diffusion term expansion ⓘ drift term expansion ⓘ multi-index notation for iterated integrals ⓘ multiple stochastic integrals ⓘ
dependsOn	moments of the driving Wiener process ⓘ regularity of drift and diffusion coefficients ⓘ
documentedIn	Numerical Solution of Stochastic Differential Equations (Kloeden and Platen) NERFINISHED ⓘ
enables	high-order strong numerical methods for SDEs ⓘ high-order weak numerical methods for SDEs ⓘ systematic derivation of stochastic Runge–Kutta schemes ⓘ
field	numerical analysis ⓘ stochastic calculus ⓘ stochastic differential equations ⓘ
generalizes	Taylor series NERFINISHED ⓘ
hasVariant	strong Itô–Taylor expansion NERFINISHED ⓘ truncated Itô–Taylor scheme ⓘ weak Itô–Taylor expansion NERFINISHED ⓘ
purpose	derive higher-order numerical schemes for SDEs ⓘ express solutions of stochastic differential equations as series ⓘ obtain strong approximations of SDE solutions ⓘ obtain weak approximations of SDE solutions ⓘ
relatedTo	Euler–Maruyama scheme NERFINISHED ⓘ Itô’s lemma NERFINISHED ⓘ Kloeden–Platen methods NERFINISHED ⓘ Milstein scheme NERFINISHED ⓘ stochastic Runge–Kutta methods NERFINISHED ⓘ stochastic Taylor formula ⓘ
typicalReference	Eckhard Platen NERFINISHED ⓘ Peter E. Kloeden NERFINISHED ⓘ
usedIn	computational finance ⓘ engineering models with noise ⓘ numerical simulation of stochastic differential equations ⓘ stochastic modeling in biology ⓘ stochastic modeling in physics ⓘ
uses	iterated Itô integrals ⓘ

Referenced by (1)

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Milstein method → relatedConcept → Itô–Taylor expansion ⓘ