An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise

Acquistapace, Paolo; Terreni, Brunello

doi:10.1080/07362998408809031

Cited by 48 publications

(58 citation statements)

References 13 publications

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“…In the infinite-dimensional case, some generalizations are known where the Wiener process is one-dimensional and the state space is infinite-dimensional (Aquistapace and Terreni [2], Brze~.niak, Capifiski and Flandoli [14], Da Prato [23], Doss [26], Gy/Sngy [33][34][35]). …”

Section: Wong-zakai Approximations In Infinite Dimensions For the Linmentioning

confidence: 99%

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Wong-Zakai approximations for stochastic differential equations

Twardowska

1996

Acta Appl Math

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The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Wong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite-and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the NavierStokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.Mathematics Subject Classifications (1991): 60H15, 60H10, 60H05, 34K50, 41A10, 65C20, 35Q30, 35R60.

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Section: Wong-zakai Approximations In Infinite Dimensions For the Linmentioning

confidence: 99%

“…In [2], the following result is stated: THEOREM 3.1 (Aquistapace and Terreni [2]). Let (f~, :T, P) be a probability space.…”

Section: Wong-zakai Approximations In Infinite Dimensions For the Linmentioning

confidence: 99%

Wong-Zakai approximations for stochastic differential equations

Twardowska

1996

Acta Appl Math

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“…Among the earliest one should mention papers by P. Acquistapace and B. Terreni, [1], Z. Brzezniak, M. Capinski, and F. Flandoli [6], I. Gyöngy [9], and Gyöngy and T. Pröhle [13]. Important recent contributions are due to V. Bally, A. Millet and M. Sanz-Solé [5], I. Gyöngy, D. Nualart and M. Sanz-Solé [12], A. Millet and M. SanzSolé [22], [23] and [24].…”

Section: Dx(t) = (Ax(t) + F (X(t)))dt + G(x(t))dw (T) (113)mentioning

confidence: 99%

“…1 Typical examples to which the theory developed in the paper is applicable are non-linear, stochastic heat equations: ∂u ∂t (t, ξ) = ∆ ξ u(t, ξ) + f (u(t, ξ)) + g(u(t, ξ)) ∂β ∂t , (1.6) u(0, ξ) = x(ξ), ξ ∈ O, u(t, ξ) = 0, t > 0, ξ ∈ ∂O, (1. 7) and strongly damped, non-linear, stochastic wave equations: ∂ 2 u ∂t 2 (t, ξ) = ∆ ξ u(t, ξ) + ρ∆ ξ ∂u ∂t (t, ξ) + f (u(t, ξ)) + g(u(t, ξ)) ∂β ∂t (1.8) u(t, ξ) = 0, t > 0, ξ ∈ ∂O, (1.9) u(0, ξ) = x 0 (ξ), ∂u ∂t (0, ξ) = x 1 (ξ), ξ ∈ O.…”

Section: Introductionmentioning

confidence: 99%

Wong-Zakai approximations of stochastic evolution equations

Tessitore

Zabczyk

2006

J. evol. equ.

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Abstract. Theorems on weak convergence of the laws of the Wong-Zakai approximations for evolution equationare proved. The operator A in the equation generates an analytic semigroup of linear operators on a Hilbert space H. The tightness of the approximating sequence is established using the stochastic factorization formula. Applications to strongly damped wave and plate equations as well as to stochastic invariance are discussed.

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“…See, for example, [1]- [9], [12]- [14], [17]- [18], and the references therein. With the exception of [4], [14], [12] and [17] the papers above prove convergence results of Wong-Zakai approximations for stochastic PDEs with various generalities, but do not present rate of convergence estimates.…”

Section: Introductionmentioning

confidence: 99%

Rate of Convergence of Wong–Zakai Approximations for Stochastic Partial Differential Equations

Gyöngy¹,

Stinga²

2013

Seminar on Stochastic Analysis, Random Fields and Applications VII

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Abstract. In this paper we show that the rate of convergence of Wong-Zakai approximations for stochastic partial differential equations driven by Wiener processes is essentially the same as the rate of convergence of the driving processes Wn approximating the Wiener process, provided the area processes of Wn also converge to those of W with that rate. We consider non-degenerate and also degenerate stochastic PDEs with time dependent coefficients.

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An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise

Cited by 48 publications

References 13 publications

Wong-Zakai approximations for stochastic differential equations

Wong-Zakai approximations for stochastic differential equations

Wong-Zakai approximations of stochastic evolution equations

Rate of Convergence of Wong–Zakai Approximations for Stochastic Partial Differential Equations

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