Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Johanna Dettweiler; J.M.A.M. van Neerven

Displaying similar documents to “Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion”

Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven, L. Weis (2005)

Studia Mathematica

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Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process ${W_{H} (t)}_{t \in [0, T]}$ . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator...

A note on maximal inequality for stochastic convolutions

Erika Hausenblas, Jan Seidler (2001)

Czechoslovak Mathematical Journal

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Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution $\int_{0}^{t} S (t - s) ψ (s) d W (s)$ driven by a Wiener process $W$ in a Hilbert space in the case when the semigroup $S (t)$ is of contraction type.

Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s, t} \in φ_{s, t} + \int_{0}^{s} \int_{0}^{t} F_{u, v} (z_{u, v}) d u d v + \int_{0}^{s} \int_{0}^{t} G_{u, v} (z_{u, v}) d w_{u, v}$

Stochastic affine evolution equations with multiplicative fractional noise

Bohdan Maslowski, J. Šnupárková (2018)

Applications of Mathematics

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A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than $1 / 2$ . Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.

A note on one-dimensional stochastic equations

Hans-Jürgen Engelbert (2001)

Czechoslovak Mathematical Journal

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We consider the stochastic equation $X_{t} = x_{0} + \int_{0}^{t} b (u, X_{u}) d B_{u}, t \geq 0,$ where $B$ is a one-dimensional Brownian motion, $x_{0} \in ℝ$ is the initial value, and $b [0, \infty) \times ℝ \to ℝ$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$ , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution. ...