A Wong–Zakai Type Approximation for Multiple Wiener–Stratonovich Integrals

Jeon, Jong Woo; Lee, Kyu Seok; Kim, Yoon Tae

doi:10.1081/sap-200026460

Cited by 9 publications

(8 citation statements)

References 7 publications

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“…Precisely, we shall prove the following: Proposition 5.1. Let (X ε , X 2,ε ) be the approximation process defined by (17) and (18). Then…”

Section: Fdd Convergence In Theorem 31mentioning

confidence: 99%

“…Let X ε and X 2,ε be defined respectively by (17) and (18). If, for every η > 0, there exists γ > µ and A < ∞ such that…”

Section: Tightness Inmentioning

confidence: 99%

“…This kind of result, usually known as diffusion approximation, has been thoroughly studied in the literature (see e.g. [17,32,33]), since it also shows that equations like (1) may emerge as the limit of a noisy equation driven by a fast oscillating function. The diffusion approximation program has also been taken up in the fBm case by Marty in [23], with some random wave problems in mind, but only in the cases where H > 1/2 or the dimension d of the fBm is 1.…”

Section: Introductionmentioning

confidence: 98%

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Weak approximation of a fractional SDE

Bardina

Nourdin²,

Rovira³

et al. 2010

Stochastic Processes and their Applications

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32 pages; this is a major revision, with two additional co-authors (X. Bardina and C. Rovira)International audienceIn this note, a diffusion approximation result is shown for stochastic differential equations driven by a (Liouville) fractional Brownian motion B with Hurst parameter H in (1/3,1/2). More precisely, we resort to the Kac-Stroock type approximation using a Poisson process studied in Bardina, Jolis and Tudor (2003) and Delgado and Jolis (2000), and our method of proof relies on the algebraic integration theory introduced by Gubinelli (2004)

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“…Precisely, we shall prove the following: Proposition 5.1. Let (X ε , X 2,ε ) be the approximation process defined by (17) and (18). Then…”

Section: Fdd Convergence In Theorem 31mentioning

confidence: 99%

“…Let X ε and X 2,ε be defined respectively by (17) and (18). If, for every η > 0, there exists γ > µ and A < ∞ such that…”

Section: Tightness Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Weak approximation of a fractional SDE

Bardina

Nourdin²,

Rovira³

et al. 2010

Stochastic Processes and their Applications

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show abstract

“…Strong convergence results are obtained in the works [6,15] for multiple Stratonovich integrals with respect to the Brownian motion and recently [24] for multiple Stratonovich fractional integrals. In the above mentioned papers the approximations are based on one of the approximations: explicit series expansion, Wong-Zakai and mollifier.…”

Section: Introductionmentioning

confidence: 88%

Multiple sub-fractional integrals and some approximations

Tudor

2008

Applicable Analysis

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We study multiple fractional integrals with respect to the even fractional Brownian motion (also called sub-fractional Brownian motion). The multiple integrals are introduced by using a representation formula for the even fractional Brownian motion as a Wiener integral with respect to a Brownian motion defined on the same probability space and a transfer principle. Then, Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals are also considered. For two standard approximations (Wong-Zakai and mollifier approximations) and continuous integrands, the mean square convergence in the uniform norm of these approximations to the multiple Stratonovich sub-fractional integral is shown.

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“…Strong approximations in mean square of multiple Stratonovich integrals are given in [3,8] for the case of Brownian motion and [17,18] for the case of cfBm.…”

Section: Introductionmentioning

confidence: 99%

A strong approximation for double subfractional integrals

Tudor

2007

Applicable Analysis

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A Riemann-Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.

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A Wong–Zakai Type Approximation for Multiple Wiener–Stratonovich Integrals

Cited by 9 publications

References 7 publications

Weak approximation of a fractional SDE

Weak approximation of a fractional SDE

Multiple sub-fractional integrals and some approximations

A strong approximation for double subfractional integrals

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