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 $z_{s, t} \in φ_{s, t} + \int_{0}^{s} \int_{0}^{t} F_{u, v} (z_{u, v}) d u d v + \int_{0}^{s} \int_{0}^{t} G_{u, v} (z_{u, v}) d w_{u, v}$

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. ...

Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław Brzeźniak, Jan van Neerven, Donna Salopek (2012)

Czechoslovak Mathematical Journal

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Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ (H, E)$ -valued functions with respect to $H$ -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0 < β < 1$ . For $0 < β < \frac{1}{2}$ we show that a function $Φ : (0, T) \to ℒ (H, E)$ is stochastically integrable with respect to an $H$ -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$ -cylindrical fractional Brownian motion. We apply our results to stochastic...

Some applications of Girsanov's theorem to the theory of stochastic differential inclusions

Micha Kisielewicz (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The Girsanov's theorem is useful as well in the general theory of stochastic analysis as well in its applications. We show here that it can be also applied to the theory of stochastic differential inclusions. In particular, we obtain some special properties of sets of weak solutions to some type of these inclusions.

Viability theorems for stochastic inclusions

Michał Kisielewicz (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Sufficient conditions for the existence of solutions to stochastic inclusions $x_{t} - x_{s} \in \int_{s}^{t} F_{τ} (x_{τ}) d τ + \int_{s}^{t} G_{τ} (x_{τ}) d w_{τ} + \int_{s}^{t} \int_{I R ⁿ} H_{τ, z} (x_{τ}) ν ̃ (d τ, d z)$ beloning to a given set K of n-dimensional cádlág processes are given.

Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's

Michal Vyoral (2005)

Applications of Mathematics

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We consider a stochastic process $X_{t}^{x}$ which solves an equation $d X_{t}^{x} = A X_{t}^{x} d t + Φ d B_{t}^{H}, X_{0}^{x} = x$ where $A$ and $Φ$ are real matrices and $B^{H}$ is a fractional Brownian motion with Hurst parameter $H \in (1 / 2, 1)$ . The Kolmogorov backward equation for the function $u (t, x) = 𝔼 f (X_{t}^{x})$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.