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Dive into the research topics where Andreas Neuenkirch is active.

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Featured researches published by Andreas Neuenkirch.


Numerische Mathematik | 2009

Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients

Arnulf Jentzen; Peter E. Kloeden; Andreas Neuenkirch

We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô–Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified Itô–Taylor scheme of order γ has pathwise convergence order γ − ε for arbitrary ε >xa00 as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.


Journal of Computational and Applied Mathematics | 2011

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

Peter E. Kloeden; Gabriel J. Lord; Andreas Neuenkirch; Tony Shardlow

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).


Annals of Operations Research | 2011

Multilevel Monte Carlo for stochastic differential equations with additive fractional noise

Peter E. Kloeden; Andreas Neuenkirch; Raffaella Pavani

We adopt the multilevel Monte Carlo method introduced by M.xa0Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order ε with a computational effort of order ε−2.


Annales De L Institut Henri Poincare-probabilites Et Statistiques | 2009

Trees and asymptotic expansions for fractional stochastic differential equations

Andreas Neuenkirch; Ivan Nourdin; Andreas Rößler; Samy Tindel

In this article, we consider an n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameterH >1/3. We derive an expansion for E[f (Xt )] in terms of t, where X denotes the solution to the SDE and f :Rn →R is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H >1/2.


Journal of Difference Equations and Applications | 2009

Synchronization of noisy dissipative systems under discretization

Peter E. Kloeden; Andreas Neuenkirch; Raffaella Pavani

It is shown that the synchronization of noisy dissipative systems is preserved when a drift-implicit Euler scheme is used for the discretization. In particular, in this case the order of discretization and synchronization can be exchanged.


Journal of Theoretical Probability | 2007

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

Andreas Neuenkirch; Ivan Nourdin


Electronic Journal of Probability | 2008

Delay equations driven by rough paths

Andreas Neuenkirch; Ivan Nourdin; Samy Tindel


Applied Mathematics and Optimization | 2009

Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion

María J. Garrido-Atienza; Peter E. Kloeden; Andreas Neuenkirch


Stochastic Processes and their Applications | 2008

Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

Andreas Neuenkirch


Stochastic Processes and their Applications | 2010

Discretizing the Fractional Lévy Area

Andreas Neuenkirch; Samy Tindel; Jérémie Unterberger

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Peter E. Kloeden

Goethe University Frankfurt

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Ivan Nourdin

University of Luxembourg

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Arnulf Jentzen

Goethe University Frankfurt

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A. Jentzen

Goethe University Frankfurt

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Andreas Rößler

Technische Universität Darmstadt

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Tony Shardlow

University of Manchester

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