Wiener Chaos and Nonlinear Filtering

Gyöngy, István; Shmatkov, A. M.

doi:10.1007/s00245-006-0873-2

Cited by 35 publications

(5 citation statements)

References 8 publications

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“…& Remark 4.5. Gyöngy & Shmatkov (2006) show a strong Wong-Zakai-type approximation result, where they also need to impose smoothness assumptions on the initial value x. Otherwise, the assumptions in Gyöngy & Shmatkov (2006) are different from ours.…”

Section: The Case When a Is Unbounded Linearmentioning

confidence: 87%

“…Gyöngy & Shmatkov (2006) show a strong Wong-Zakai-type approximation result, where they also need to impose smoothness assumptions on the initial value x. Otherwise, the assumptions in Gyöngy & Shmatkov (2006) are different from ours. They allow linear, densely defined vector fields and general adapted coefficients; on the other hand, the generator A needs to be elliptic.…”

Section: The Case When a Is Unbounded Linearmentioning

confidence: 87%

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Cubature on Wiener space in infinite dimension

Bayer

Teichmann

2008

Proc. R. Soc. A.

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We prove a stochastic Taylor expansion for stochastic partial differential equations (SPDEs) and apply this result to obtain cubature methods, i.e. high-order weak approximation schemes for SPDEs, in the spirit of Lyons and Victoir (Lyons & Victoir 2004 Proc. R. Soc. A 460, 169-198). We can prove a high-order weak convergence for welldefined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of Lévy processes of finite type; here the results seem to be new, even in finite dimension. Several numerical examples are added.

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Section: The Case When a Is Unbounded Linearmentioning

confidence: 87%

Section: The Case When a Is Unbounded Linearmentioning

confidence: 87%

Cubature on Wiener space in infinite dimension

Bayer

Teichmann

2008

Proc. R. Soc. A.

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“…On the other hand, since u δ,τ −t ∈ D(τ − t, θ t ω) and D ∈ D, by (29) we find that there exists T = T (τ, ω, D, δ) > 0 such that for all t ≥ T ,…”

mentioning

confidence: 79%

“…Since νλ > α, by (29), there exists s 0 < 0 such that for all s ≤ s 0 , s 0 (νλ − 2βG δ (θ r ω))dr < αs.…”

Section: 2mentioning

confidence: 95%

Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations

Gu¹,

Lu²,

Wang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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185 186 ANHUI GU, KENING LU AND BIXIANG WANGWe are interested in the existence of random attractors for equation (1) when R is a nonlinear Lipschitz continuous function, for which we need to define a random dynamical system via the solution operators because the attractors theory of stochastic equations is formulated in terms of random dynamical systems. However, for a general Lipschitz nonlinearity R, the existence of such a random dynamical system for (1) is unknown (see, e.g., [22]), and hence the random attractors theory does not apply in this case directly. As far as the authors are aware, the existence of random attractors for the autonomous version of (1) was proved only for a special form of R and is still open for a general Lipschitz diffusion term (see, e.g., [8,18,21]). In order to solve this problem, in the present paper, we propose to study the pathwise dynamics of the stochastic equation (1) by the Wong-Zakai approximations defined as a stationary process given by the Wiener shift (see Section 2 for details). Indeed, as we will see later, the corresponding random equations driven by the Wong-Zakai approximations generate a random dynamical system for a general Lipschitz diffusion term R, and have a unique tempered random attractor (see Theorem 2.3). This indicates that random equations with pathwise approximations are more amenable to analysis than stochastic equations driven by white noise, as far as pathwise dynamics is concerned. Furthermore, we will prove the solutions of the approximate equations converge to that of the corresponding stochastic equations driven by linear multiplicative noise or additive noise when the step size of the approximations tends to zero. The continuity of random attractors for the approximate equations will also be established in these cases.The Wong-Zakai approximations were first proposed in [61,62] where the authors developed the idea of using pathwise deterministic equations to approximate stochastic ones driven by one-dimensional Brownian motions. Currently, the Wong-Zakai approximations have been extended to higher-dimensional Brownian motions as well as martingales and semimartingales, see, e.g.], and the references therein. In the present paper, we will use the idea of Wong-Zakai approximations to study the existence and uniqueness of tempered random attractors of (1). Such attractors have been extensively studied in the literature, see [4,5,9,11,12,13,17,18,21,24,25,26,27,33,37,49,51,57] for autonomous stochastic equations; and [14, 20, 58, 59, 60] for non-autonomous stochastic equations. In this paper, we will deal with the non-autonomous stochastic equations (1).In the next section, we prove the existence and uniqueness of random attractors for the Navier-Stokes equations driven by the Wong-Zakai approximations. In the last two sections, we prove the convergence of solutions and attractors of the approximate equations when the step size of approximations approaches zero, for linear multiplicative noise and additive noise, respectively. 2.Random attractors ...

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“…There are also a lots of publications on Wong-Zakai approximations of solutions for stochastic partial differential equations, see for example, Brzezniak et al [4], Gyongy [20,18], Twardowska [52,51,50,53], Bally et al [2], Brzezniak and Flandoli [5], Grecksch and Schmalfus [16], Gyongy and Shmatkov [20], Nowak [40], Tessitore and Zabczyk [49], Deya et al [8], Ganguly [13], and Hairer and Pardoux [21].…”

mentioning

confidence: 99%

Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

Ma¹,

Shu²,

Qin³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we discuss the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for stochastic Ginzburg-Landau equations driven by a white noise. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions. Consequently, we show that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. At last, when the stochastic Ginzburg-Landau equation is driven by a linear multiplicative noise, we establish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.2010 Mathematics Subject Classification. Primary: 37L55, 60H15; Secondary: 35Q56.

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Wiener Chaos and Nonlinear Filtering

Cited by 35 publications

References 8 publications

Cubature on Wiener space in infinite dimension

Cubature on Wiener space in infinite dimension

Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations

Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

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