Displaying similar documents to “Stability of semilinear equations with boundary and pointwise noise”

Invariant measures for nonlinear SPDE's: uniqueness and stability

Bohdan Maslowski, Jan Seidler (1998)

Archivum Mathematicum

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The paper presents a review of some recent results on uniqueness of invariant measures for stochastic differential equations in infinite-dimensional state spaces, with particular attention paid to stochastic partial differential equations. Related results on asymptotic behaviour of solutions like ergodic theorems and convergence of probability laws of solutions in strong and weak topologies are also reviewed.

Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process

Zdzisław Brzeźniak, Szymon Peszat (1999)

Studia Mathematica

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Stochastic partial differential equations on ℝ d are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted L q -space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.

SPDEs with pseudodifferential generators: the existence of a density

Samy Tindel (2000)

Applicationes Mathematicae

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We consider the equation du(t,x)=Lu(t,x)+b(u(t,x))dtdx+σ(u(t,x))dW(t,x) where t belongs to a real interval [0,T], x belongs to an open (not necessarily bounded) domain 𝒪 , and L is a pseudodifferential operator. We show that under sufficient smoothness and nondegeneracy conditions on L, the law of the solution u(t,x) at a fixed point ( t , x ) ∈ [ 0 , T ] × 𝒪 is absolutely continuous with respect to the Lebesgue measure.

Stochastic invariance and consistency of financial models

Jerzy Zabczyk (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The paper is devoted to a connection between stochastic invariance in infinite dimensions and a consistency question of mathematical finance. We derive necessary and sufficient conditions for stochastic invariance of Nagumo’s type for stochastic equations with additive noise. They are applied to Ornstein-Uhlenbeck processes and to specific financial models. The case of evolution equations with general noise is discussed also and a comparison with recent results obtained by geometric...