Quasi-Sure Convergence Rate of Euler Scheme for Stochastic Differential Equations

Wenliang Huang; Xicheng Zhang

doi:10.1016/S0252-9602(13)60126-5

Quasi-Sure Convergence Rate of Euler Scheme for Stochastic Differential Equations

Wenliang Huang^*, Xicheng Zhang

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let X_t(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xⁿ_t(x) be the Euler discretization scheme of SDEs with step 2^-n. In this note, we prove that for any R > 0 and γ ∈ (0,1/2), supt∈[0,1],|x|≤R|Xtn(x,ω)-Xt(x,ω)|≤ξR,γ(ω)2 nγ,n≥1,q.e.,where ξR,γ(ω) is quasi-everywhere finite.

Original language	English
Pages (from-to)	65-72
Number of pages	8
Journal	Acta Mathematica Scientia
Volume	34
Issue number	1
DOIs	https://doi.org/10.1016/S0252-9602(13)60126-5
Publication status	Published - Jan 2014
Externally published	Yes

Keywords

Euler approximation
Quasi-sure convergence
SDE

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10.1016/S0252-9602(13)60126-5

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abstract = "Let Xt(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt(x) be the Euler discretization scheme of SDEs with step 2-n. In this note, we prove that for any R > 0 and γ ∈ (0,1/2), supt∈[0,1],|x|≤R|Xtn(x,ω)-Xt(x,ω)|≤ξR,γ(ω)2 nγ,n≥1,q.e.,where ξR,γ(ω) is quasi-everywhere finite.",

keywords = "Euler approximation, Quasi-sure convergence, SDE",

author = "Wenliang Huang and Xicheng Zhang",

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N2 - Let Xt(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt(x) be the Euler discretization scheme of SDEs with step 2-n. In this note, we prove that for any R > 0 and γ ∈ (0,1/2), supt∈[0,1],|x|≤R|Xtn(x,ω)-Xt(x,ω)|≤ξR,γ(ω)2 nγ,n≥1,q.e.,where ξR,γ(ω) is quasi-everywhere finite.

AB - Let Xt(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt(x) be the Euler discretization scheme of SDEs with step 2-n. In this note, we prove that for any R > 0 and γ ∈ (0,1/2), supt∈[0,1],|x|≤R|Xtn(x,ω)-Xt(x,ω)|≤ξR,γ(ω)2 nγ,n≥1,q.e.,where ξR,γ(ω) is quasi-everywhere finite.

KW - Euler approximation

KW - Quasi-sure convergence

KW - SDE

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EP - 72

JO - Acta Mathematica Scientia

JF - Acta Mathematica Scientia

IS - 1

ER -

Quasi-Sure Convergence Rate of Euler Scheme for Stochastic Differential Equations

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Keywords

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