Some results on semilinear stochastic differential equations in hilbert spaces

Prato, D. Da; Tubaro, Luciano

doi:10.1080/17442508508833360

Cited by 33 publications

(48 citation statements)

References 3 publications

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“…These results have appeared in the literature [1], [5], [18] in the general case of infinite-dimensional Wiener processes. Here we will consider only the finitedimensional case, which is the interesting one for nonlinear filtering.…”

Section: Preliminaries On Mild Equationsmentioning

confidence: 52%

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

Germani

Piccioni

1987

Appl Math Optim

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Abstract. The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitable L 2 space. It is assumed that the Markov semigroup corresponding to the state process is Co on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.

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Section: Preliminaries On Mild Equationsmentioning

confidence: 52%

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

Germani

Piccioni

1987

Appl Math Optim

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“…The techniques of [11] cannot be extended to the setting of martingale type 2 spaces E, since this would require an extension of the Itô formula for the duality mapping. Here the problem arises that if E has martingale type 2, then E * has martingale type 2 only if E is isomorphic to a Hilbert space (see [21,28]).…”

Section: Du (T) = A(t)u (T) Dt + Bu (T) Dw (T) T ∈ [0 T ]mentioning

confidence: 98%

“…The following condition is introduced in [13, Theorem 6.30] (see also [11,Proposition 1]) in the time independent case. Let (C (t)) t∈[0,T ] be densely defined and such that 0 ∈ (C (t)) for all t ∈ [0, T ].…”

Section: The Deterministic Problem: Acquistapace-terreni Conditionsmentioning

confidence: 99%

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Brzeźniak

Neerven

Veraar

et al. 2008

Journal of Differential Equations

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Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. (Z. Brzeźniak), j.m.a.m.vanneerven@tudelft.nl (J.M.A.M. van Neerven), m.c.veraar@tudelft.nl (M.C. Veraar), lutz.weis@mathematik.uni-karlsruhe.de (L. Weis).

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“…It is well known that these topics have been developed mainly by using two different methods, that is, the semigroup approach (for example, Chojnowska-Michalik [4], Da Prato et al [7], Dawson [11] and Kotelenez [14]) and the variational one (for example, Krylov and Rozovskii [15] and Pardoux [19]). On the other hand, although stochastic partial functional differential equations with finite delays also seem very important as stochastic models of biological, chemical, physical and economical systems, the corresponding properties of these systems have not been studied in great detail (cf.…”

Section: Introductionmentioning

confidence: 99%

Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

Taniguchi¹,

Li²,

Truman³

2002

Journal of Differential Equations

171

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In this paper we shall consider the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r > 0: dX(t)=[ − AX(t)+f(t, X t )] dt+g(t, X t ) dW(t), where we assume that − A is a closed, densely defined linear operator and the generator of a certain analytic semigroup. f:is a given K-valued Wiener process and both H and K are separable Hilbert spaces.

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Some results on semilinear stochastic differential equations in hilbert spaces

Cited by 33 publications

References 3 publications

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

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