Markus Riedle
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View article: Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces
Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces Open
In this work, we present a comprehensive theory of stochastic integration with respect to arbitrary cylindrical Lévy processes in Hilbert spaces. As cylindrical Lévy processes do not enjoy a semimartingale decomposition, our approach relie…
View article: Regularisation of cylindrical Lévy processes in Besov spaces
Regularisation of cylindrical Lévy processes in Besov spaces Open
In this work, we quantify the irregularity of a given cylindrical Lévy process $L$ in $L^2({\mathbb R}^d)$ by determining the range of weighted Besov spaces $B$ in which $L$ has a regularised version $Y$, that is a stochastic process $Y$ i…
View article: Lévy measures on Banach spaces
Lévy measures on Banach spaces Open
We establish an explicit characterisation of Lévy measures on both $L^p$-spaces and UMD Banach spaces. In the case of $L^p$-spaces, Lévy measures are characterised by an integrability condition, which directly generalises the known descrip…
View article: Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces
Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces Open
In this work, we present a comprehensive theory of stochastic integration with respect to arbitrary cylindrical Lévy processes in Hilbert spaces. Since cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach r…
View article: SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula
SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula Open
We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $α$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetr…
View article: Stochastic integration with respect to canonical $α$-stable cylindrical Lévy processes
Stochastic integration with respect to canonical $α$-stable cylindrical Lévy processes Open
In this work, we introduce a theory of stochastic integration with respect to symmetric $α$-stable cylindrical Lévy processes. Since $α$-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based …
View article: Stochastic evolution equations driven by cylindrical stable noise
Stochastic evolution equations driven by cylindrical stable noise Open
We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard α-stable cylindrical Lévy process defined on a Hilbert space for α∈(1,2). The coefficients are assumed to map between certain doma…
View article: Stochastic integration with respect to canonical α-stable cylindrical Lévy processes
Stochastic integration with respect to canonical α-stable cylindrical Lévy processes Open
In this work, we introduce a theory of stochastic integration with respect to symmetric α-stable cylindrical Lévy processes. Since α-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a…
View article: Stochastic evolution equations driven by cylindrical stable noise
Stochastic evolution equations driven by cylindrical stable noise Open
We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $α$-stable cylindrical Lévy process defined on a Hilbert space for $α\in (1,2)$. The coefficients are assumed to map between certa…
View article: Variational solutions of stochastic partial differential equations with cylindrical Lévy noise
Variational solutions of stochastic partial differential equations with cylindrical Lévy noise Open
In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t)) \dL(t)$$ driven by a cylindrical Lévy process $L$ is established. The coefficients $F$ an…
View article: The stochastic Cauchy problem driven by a cylindrical Lévy process
The stochastic Cauchy problem driven by a cylindrical Lévy process Open
In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stocha…
View article: Stochastic integration with respect to cylindrical L\\'evy processes by\n p-summing operators
Stochastic integration with respect to cylindrical L\\'evy processes by\n p-summing operators Open
We introduce a stochastic integral with respect to cylindrical L\\'evy\nprocesses with finite $p$-th weak moment for $p\\in [1,2]$. The space of\nintegrands consists of $p$-summing operators between Banach spaces of\nmartingale type $p$. W…
View article: Variational solutions of stochastic partial differential equations with\n cylindrical L\\'evy noise
Variational solutions of stochastic partial differential equations with\n cylindrical L\\'evy noise Open
In this article, the existence of a unique solution in the variational\napproach of the stochastic evolution equation $$\\dX(t) = F(X(t)) \\dt + G(X(t))\n\\dL(t)$$ driven by a cylindrical L\\'evy process $L$ is established. The\ncoefficien…
View article: Stable cylindrical Lévy processes and the stochastic Cauchy problem
Stable cylindrical Lévy processes and the stochastic Cauchy problem Open
In this work, we consider the stochastic Cauchy problem driven by the canonical $α$-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficien…
View article: Large deviations for stochastic heat equations with memory driven by Lévy-type noise
Large deviations for stochastic heat equations with memory driven by Lévy-type noise Open
For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with mem…
View article: Stochastic integration with respect to cylindrical Lévy processes
Stochastic integration with respect to cylindrical Lévy processes Open
A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integr…
View article: Stochastic integration with respect to cylindrical Lévy processes
Stochastic integration with respect to cylindrical Lévy processes Open
A cylindrical Lévy process does not enjoy a cylindrical version of thesemimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral…
View article: Copulas in Hilbert spaces
Copulas in Hilbert spaces Open
In this article, the concept of copulas is generalised to infinite\ndimensional Hilbert spaces. We show one direction of Sklar's theorem and\nexplain that the other direction fails in infinite dimensional Hilbert spaces.\nWe derive a neces…
View article: Copulas in Hilbert spaces
Copulas in Hilbert spaces Open
In this article, the concept of copulas is generalised to infinite dimensional Hilbert spaces. We show one direction of Sklar's theorem and explain that the other direction fails in infinite dimensional Hilbert spaces. We derive a necessar…
View article: Radonifying operators and infinitely divisible Wiener integrals
Radonifying operators and infinitely divisible Wiener integrals Open
In this article we illustrate the relation between the existence of Wiener integrals with respect to a Levy process in a separable Banach space and radonifying operators. For this purpose, we introduce the class of theta-radonifying operat…
View article: Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces
Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces Open
We study the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type ∫Fγ(t−s)dL(s) to a process ∫F(t−s)dL(s) driven by a Lévy process L. In Banach spaces we in…
View article: Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes
Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes Open
In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Lévy processes in separable Banach spaces. Here, a cylindrical Lévy process is understood in the classical frame…