Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

Bao, Jianhai; Huang, Xing; Chen, Yuan

doi:10.1007/s10959-018-0854-9

Cited by 40 publications

(42 citation statements)

References 26 publications

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“…Herein, G and b are given as in (2.1), g : We assume that b and G are such that (A1) and (A2) hold with σ ≡ 0 n×m therein. We further suppose that there exist L 0 , r > 0 such that for any x, y, x, y ∈ R n and u ∈ U, By following the procedures of (2.2) and (2.3), the discrete-time EM scheme and the continuous-time EM approximation associated with (3.1) are defined respectively as follows: 2) where N nh := N((n + 1)h, U) -N(nh, U), and…”

Section: The Nsdde Driven By Pure Jump Processesmentioning

confidence: 99%

“…For example, under the Khasminskii-type condition, Mao [11] revealed that the convergence rate of the truncated EM method is close to one-half; under the Hölder condition, the convergence rate of EM scheme for SDEs has been studied by many scholars (see, e.g., [7,16,17]); Sabanis [19] recovered the classical rate of convergence (i.e., one-half ) for the tamed EM schemes, where, for the SDE involved, the drift coefficient satisfies a onesided Lipschitz condition and a polynomial Lipschitz condition, and the diffusion term is Lipschitzian. In [2], Bao et al investigated the convergence rate of EM scheme for SDEs with Hölder-Dini continuous drifts.…”

Section: Introductionmentioning

confidence: 99%

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Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

2021

J Inequal Appl

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In this paper, we are concerned with the convergence rate of Euler–Maruyama (EM) scheme for stochastic differential delay equations (SDDEs) of neutral type, where the neutral, drift, and diffusion terms are allowed to be of polynomial growth. More precisely, for SDDEs of neutral type driven by Brownian motions, we reveal that the convergence rate of the corresponding EM scheme is one-half; Whereas for SDDEs of neutral type driven by pure jump processes, we show that the best convergence rate of the associated EM scheme is slower than one-half. As a result, the convergence rate of general SDDEs of neutral type, which is dominated by pure jump process, is slower than one-half.

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Section: The Nsdde Driven By Pure Jump Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

2021

J Inequal Appl

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“…To avoid complicated computation, in the present setup we work only on the case that the drift is uniformly bounded. Nevertheless, employing the standard cut-off approach (see, e.g., [1]), we of course can extend our framework to the setting that the drift coefficient is unbounded.…”

Section: Approximation Of Semi-linear Spdes With Hölder Continuous Drmentioning

confidence: 99%

“…Assume (A1), (4). For any λ ≥ λ T , let u λ be the unique solution to (1). Then the following assertions hold.…”

mentioning

confidence: 98%

New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts

Bao¹,

Huang²,

Chen³

2019

Communications on Pure &Amp; Applied Analysis

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In this paper, some new results on the the regularity of Kolmogorov equations associated to the infinite dimensional OU-process are obtained. As an application, the average L 2 -error on [0, T ] of exponential integrator scheme for a range of semi-linear stochastic partial differential equations is derived, where the drift term is assumed to be Hölder continuous with respect to the Sobolev norm · β for some appropriate β > 0. In addition, under a stronger condition on the drift, the strong convergence estimate is obtained, which covers the result of the SDEs with Hölder continuous drift.2000 Mathematics Subject Classification. Primary: 60H15, 35R60, 65C30.

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“…Meanwhile, in order to understand the numerical approximation of SDEs with irregular coe cients, numerical schemes have been established. The strong and weak convergence rates of EM scheme for SDEs with singular coe cients have already been obtained, see for instance [4,5,38,45,46,48,49,50,76,79,80,89]. The references [23,51,71,72,73,77,84] investigated L p -approximation of solutions to the SDEs with singular drift, and obtained the corresponding L p -error rates under the di↵erential assumptions about the coe cients.…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

Suo¹

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In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we ﬁrst construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we ﬁrst establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model.

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Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

Cited by 40 publications

References 26 publications

Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

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