Markus Riedle

Mean square stability of stochastic Volterra integro-differential equations

Abstract The mean square stability of a non-linear stochastic Volterra integro-differential equation is studied. Non-convolution Volterra terms arise in both the drift and the dispersion term. Moreover, for the convolution case we determine the rate of convergence in terms of an integrability condition on the Volterra kernels.

Cylindrical Lévy processes in Banach spaces

Cylindrical probability measures are flnitely 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-It^o decompositions and an associated Levy- Khintchine formula. If the process is weakly square integrable, its covariance oper- ator can be used to construct a reproducing kernel Hilbert space in which the pro- cess has a decomposition as an inflnite series built from a sequence of uncorrelated bona flde one-dimensional Levy processes. This series is used to deflne 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.

Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

Abstract In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.

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.

On Émery's Inequality and a Variation-of-Constants Formula

Abstract A generalization of Émerys inequality for stochastic integrals is shown for convolution integrals of the form , where Z is a semimartingale, Y an adapted càdlàg process, and g a deterministic function. An even more general inequality for processes with two parameters is proved. The inequality is used to prove existence and uniqueness of solutions of equations of variation-of-constants type. As a consequence, it is shown that the solution of a semilinear delay differential equation with functional Lipschitz diffusion coefficient and driven by a general semimartingale satisfies a variation-of-constants formula.

Bubbles and crashes in a Black–Scholes model with delay

This paper studies the asymptotic behaviour of an affine stochastic functional differential equation modelling the evolution of the cumulative return of a risky security. In the model, the traders of the security determine their investment strategy by comparing short- and long-run moving averages of the security’s returns. We show that the cumulative returns either obey the law of the iterated logarithm, but have dependent increments, or exhibit asymptotic behaviour that can be interpreted as a runaway bubble or crash.

Geometric Brownian motion with delay: mean square characterisation

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficient.

Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An L2 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.

Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces

We study the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Lévy process L. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior 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 diagonalizable generators.

Stochastic integration with respect to cylindrical Lévy processes

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 integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible integrands consists of adapted stochastic processes with values in the space of Hilbert-Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semi-martingale with cadlag trajectories.