Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven; L. Weis

Displaying similar documents to “Stochastic integration of functions with values in a Banach space”

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

Johanna Dettweiler, J.M.A.M. van Neerven (2006)

Czechoslovak Mathematical Journal

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Let $A = d / d θ$ denote the generator of the rotation group in the space $C (Γ)$ , where $Γ$ denotes the unit circle. We show that the stochastic Cauchy problem $d U (t) = A U (t) + f d b_{t}, U (0) = 0, (1)$ where $b$ is a standard Brownian motion and $f \in C (Γ)$ is fixed, has a weak solution if and only if the stochastic convolution process $t \mapsto {(f * b)}_{t}$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution...

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

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.

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.

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...

The method of lines for hyperbolic stochastic functional partial differential equations

Monika Wrzosek, Maria Ziemlańska (2018)

Czechoslovak Mathematical Journal

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We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small $L^{2}$ -perturbations.

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.

Pathwise uniqueness for stochastic PDEs

Giuseppe Da Prato (2015)

Banach Center Publications

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We consider a stochastic evolution equation in a separable Hilbert spaces H or in a separable Banach space E with a Hölder continuous perturbation on the drift. We review some recent result about pathwise uniqueness for this equation.

Uniqueness for stochastic evolution equations in Banach spaces

Martin Ondreját

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Different types of uniqueness (e.g. pathwise uniqueness, uniqueness in law, joint uniqueness in law) and existence (e.g. strong solution, martingale solution) for stochastic evolution equations driven by a Wiener process are studied and compared. We show a sufficient condition for a joint distribution of a process and a Wiener process to be a solution of a given SPDE. Equivalences between different concepts of solution are shown. An alternative approach to the construction of the stochastic...

Stochastic differential inclusions

Michał Kisielewicz (1997)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The definition and some existence theorems for stochastic differential inclusions depending only on selections theorems are given.

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.

Topological dual of $B_{\infty} (I, ₁ (X, Y))$ with application to stochastic systems on Hilbert space

N.U. Ahmed (2009)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as...