arXiv subjects

Markus Riedle

Publications and source records attributed to Markus Riedle.

26 records · Page 2Linked to original sources

arXiv2015

Radonifying operators and infinitely divisible Wiener integrals

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 operators, i.e. operators which map a cylindrical measure theta to a genuine Radon measure. We study this class of operators for various examples of infinitely divisible cylindrical measures theta and highlight the differences from the Gaussian case.

math.PR↗

arXiv2014

Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces

We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type $\int_0^t F_γ(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a Lévy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein--Uhlenbeck processes with diagonalisable generators.

math.PR↗

arXiv2014

Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes

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 framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes and several examples are considered.

math.PR↗

arXiv2013

Cylindrical Fractional Brownian Motion in Banach Spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen-Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.

math.PR↗

arXiv2012

Stochastic integration with respect to cylindrical Levy processes in Hilbert spaces: an L^2 approach

In this work stochastic integration with respect to cylindrical Levy processes with weak second moments is introduced. It is well known that a deterministic Hilbert-Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Levy process by random Hilbert-Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Levy process. We finish this work by deriving an Ito isometry and by considering shortly stochastic partial differential equations driven by cylindrical Levy processes.

math.PR↗

arXiv2011

Infinitely divisible cylindrical measures on Banach spaces

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Further properties of infinitely divisible cylindrical measures such as continuity are derived. Moreover, the result on the classification enables us to conclude new results on genuine Levy measures on Banach spaces.

math.PR↗

arXiv2009

Cylindrical Levy processes in Banach spaces

Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.

math.PR↗

arXiv2008

Cylindrical Wiener processes

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well-known stochastic integrals in literature.

math.PR↗