Weak error analysis for semilinear stochastic Volterra equations with additive noise

Andersson, Adam; Kovács, Mihály; Larsson, Stig

doi:10.1016/j.jmaa.2015.09.016

Cited by 21 publications

(60 citation statements)

References 25 publications

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“…The following strong convergence result can be proven analogously to the Gaussian case, cf. [1,Theorem 4.2]. Proposition 4.3 (Strong convergence).…”

Section: 3mentioning

confidence: 99%

“…For the weak convergence we consider path dependent functionals as specified by the next assumption. In the related work [1] functionals of the form f (x) = n i=1 ϕ i [0,T ] x(t) µ i (dt) , with ϕ 1 , . .…”

Section: 3mentioning

confidence: 99%

“…Here we prove an analogue of Proposition 3.5 for the discrete solution. It has Gaussian counterparts in [1,Proposition 4.3] and [2,Proposition 3.17]. Up to some straightforward modifications, the proof of (32) is analogous to that of [2,Proposition 3.16] in the Gaussian case and is therefore omitted.…”

Section: 2mentioning

confidence: 99%

“…In this article, we extend the strategy for semilinear SPDE from [1,2] to Poisson noise and analyze the weak approximation error in a framework of Gelfand triples of refined Malliavin-Sobolev spaces M 1,p,q (H) ⊂ L 2 (Ω; H) ⊂ (M 1,p,q (H)) * , see Subsection 3.2 for the definition of these spaces. We first investigate in Section 3 the Malliavin regularity of the mild solution X = (X(t)) t 0 to Eq.…”

Section: Introductionmentioning

confidence: 99%

“…Convergence in negative order spaces. The following crucial result has Gaussian counterparts in[1, Lemma 4.6] and[2, Lemma 4.6].…”

mentioning

confidence: 99%

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Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Andersson¹,

Lindner²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.2010 Mathematics Subject Classification. 60H15, 60G51, 60H07, 65C30, 65M60.

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“…The following strong convergence result can be proven analogously to the Gaussian case, cf. [1,Theorem 4.2]. Proposition 4.3 (Strong convergence).…”

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Convergence in negative order spaces. The following crucial result has Gaussian counterparts in[1, Lemma 4.6] and[2, Lemma 4.6].…”

mentioning

confidence: 99%

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Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Andersson¹,

Lindner²

2019

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

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Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise

Hu,

Yan,

Sarwar

2023

Numerical Methods Partial

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Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Naurois¹,

Jentzen

Welti

2021

Appl Math Optim

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Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms.

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Weak error analysis for semilinear stochastic Volterra equations with additive noise

Cited by 21 publications

References 25 publications

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise

Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

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