Stochastic (Partial) Differential Equations Day im Sommersemester 2017

The scientific program is organized by Lisa Beck (Universität Augsburg), Dirk Blömker (Universität Augburg), and Nina Gantert (TUM).

There is no conference fee. Everybody is welcome to attend.

Location and dates

Monday, 26th June 2017, starting 14:15, at TUM, lecture hall 2.02.01, Parkring 11, Garching-Hochbrück (Technische Universität München).

How to visit us at Garching-Hochbrück.

Speakers

Program, Titles, and Abstracts

We will go for dinner with the speakers after the talks.

  • 14:30 - 15:30 Andrea Barth
    Title: Approximations of Stochastic Partial Differential Equations with L\'evy noise Abstract: Empirical observations of energy forward markets suggest that the forward curves in these markets are subject to a large number of idiosyncratic risk sources, which motivates the use of an infinite-dimensional source of noise to model the dynamics in these markets. This is in contrast to interest rate derivatives, where the stochastic variation in general may be captured by a low-dimensional noise process. One approach to model energy forward dynamics is to consider first order hyperbolic stochastic partial differential equations (SPDEs), basically an extension of the Heath-Jarrow-Morton framework to infinite dimensions. We consider SPDEs where the driving noise term is a square-integrable, Hilbert space-valued L\'evy process, which allows us to model discontinuities in the forward curves and asymmetric marginal distributions in a very flexible way. However, compared to a Gaussian random field, the approximation of these L\'evy fields is not straightforward and may be computationally expensive. We use truncated Karhunen-Lo\`eve expansions to obtain a suitable finite-dimensional representation of the infinite-dimensional processes and control for the corresponding cutoff-error. To simulate the remaining finite-dimensional L\'evy processes, we develop an algorithm based on Fourier inversion techniques, show how to embed it into discretization schemes for hyperbolic SPDEs and derive mean-square convergence results and error bounds. This is joint work with Andreas Stein (SimTech, University of Stuttgart)
  • 15:40 - 16:40 Raphael Kruse
    Title:On randomized one-step methods for non-autonomous (stochastic) differential equations with time-irregular coefficients
    Abstract: In this talk we discuss the numerical solution of non-autonomous (stochastic) differential equations whose coefficient functions are non-smooth. For instance, we are interested in ODEs of Carath´eodory type, whose vector field is allowed to be discontinuous or contains a weak singularity with respect to the temporal parameter. Already in the deterministic case it is well-known that standard numerical solvers only converge very slowly to the exact solution or might even be divergent, if they only depend on finitely many point evaluations of the coefficient function. We therefore focus on certain randomization strategies for numerical one-step methods, which often offer several advantages. In particular, we will see that in the case of Carath´eodory type ODEs the order of convergence of a randomized solver is increased by 0.5 compared to its deterministic counter-part under the same regularity assumptions on the coefficient functions. In a further application we consider a randomized Milstein method for stochastic differential equations. In this case it will turn out, that we achieve the same order of convergence as the standard Milstein method but under significantly relaxed smoothness assumption on the drift coefficient function. If time permits we will finally indicate generalizations to some nonlinear evolution equations in infinite dimensions. This talk is based on joint work [1, 2] with Yue Wu. [1] R. Kruse and Y. Wu. Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients. Comput. Methods Appl. Math., 2017. (to appear). [2] R. Kruse and Y. Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Preprint, 2017
  • 17:00 - 18:00 Martina Hofmanova
    Title: Stochastic mean curvature flow
    Abstract: Motion by mean curvature of embedded hypersurfaces in $\mathbb{R}^{N+1}$ is an important prototype of a geometric evolution law and has been intensively studied in the past decades. Mean curvature flow is characterized as a steepest descent evolution for the surface area energy and constitutes a fundamental relaxation dynamics for many problems where the interface size contributes to the systems energy. One of the main difficulties of the mean curvature flow is the appearance of topological changes and singularities in finite time. Further issues then arise in the mathematical treatment of the stochastic mean curvature flow, which was introduced as a refined model incorporating the influence of thermal noise. We study a stochastically perturbed mean curvature flow for graphs in $\mathbb{R}^3$ over the two-dimensional unit-cube subject to periodic boundary conditions. In particular, we establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.