Mitteldeutscher Stochastik Workshop

Abstracts for the talks

Stefan Ankirchner
FSU Jena
Optimal Stopping with expectation constraints
We discuss several methods for solving stopping problems with expectation constraints.

Mahdi Azimi
Martin-Luther-Universität Halle
Backward stochastic Volterra integral equations in Lq spaces and duality principle with forward stochastic Volterra integral equations

We consider a backward stochastic Volterra integral equation [BSVIE] in the Banach space

E = Lq(S,Σ), where μ is σ-finte measure. The stochastic integral

is defined with respect to an infinite dimensional Wiener process. Under appropriate assumptions,

the existence and uniqueness of an adapted solution of BSVIE are being proofed by using martingale

representation theorem in Banach space E and Banach fixed-point theorem. Some properties of the solution

are also discussed. We prove a duality principle between BSVIEs and forward Itô Volterra stochastic integral

equations with respect to a cylindrical Wiener process.

Markus Böhm
FSU Jena
A lower bound on the Hausdorff dimension of random attractors
We consider an SPDE with a Lipschitz continuous nonlinearity driven by a multiplicative
noise. Its mild solution generates a random dynamical system and we discuss the exis-
tence of a random attractor. Applying a cut-off function to the nonlinearity we obtain a
local unstable manifold for the random dynamical system. The local unstable manifold
contains an open subset which lies also in the related random attractor. This subset is
used to obtain a lower bound on the Hausdorff dimension of the random attractor.

Maximilian Büttner
Martin-Luther-Universität Halle
Simulation of a solution of a kolmogorov-type equation for a fractional Ornstein-Uhlenbeck process
1. fractional Brownian motion (fBm) and an integral with respect to the fBm
2. Properties of the solution of an Ornstein-Uhlenbeck process
3. relation to partial differential equations
4. Estimation of the solution with Monte-Carlo methods
5. Introduction of Importance sampling to reduce the variance of the estimator

Benjamin Fehrman
Pathwise well-posedness of a stochastic porous medium equation with nonlinear, conservative noise
In this talk, which is based on joint work with Benjamin Gess, I will present a pathwise well-posedness theory for porous medium equations driven by conservative, nonlinear noise. The solution theory is based upon the kinetic formulation of the equation. On this level, the noise enters linearly, and the corresponding system of stochastic characteristics may be understood as rough paths. This yields a well-defined, pathwise notion of solution, for which test functions are transported along the inverse system of characteristics. I will discuss the existence and uniqueness of such solutions, and I will mention some estimates proving that, locally in time, pathwise solutions possess the basic spacial regularity of solutions to the deterministic porous medium equation.

Liang Fei
Xi'an + FSU Jena
Global existence and explosion of the stochastic viscoelastic wave equation driven by multiplicative noises
We discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term |ut|q2ut and a source term of the type |u|p2u. We firstly establish the local existence and uniqueness of solution by the iterative technique truncation function method. Moreover, we also show that the solution is global for q p. Lastly, by modifying the energy functional, we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for p > q.

Alexander Fromm
FSU Jena
Optimal position targeting via decoupling fields
We consider a variant of the basic problem of the calculus of variations, where the Lagrangian
is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by
reducing it, via a limiting argument, to an unconstrained control problem that consists in finding
an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation
of the terminal state from a given target position. Using the Pontryagin maximum principle we
characterize a solution of the unconstrained control problem in terms of a fully coupled
forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields
for proving that the FBSDE has a unique solution.

Benjamin Gess
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Path-by-path regularization by noise for scalar conservation laws
We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [G., Souganidis; Comm. Pure Appl. Math. (2017)]. We show that (ρ,γ)-regularity is a sufficient path-by-path condition implying such regularizing effects. In addition, we introduce a new path-by-path scaling property which is also shown to be sufficient to imply regularizing effects.

Wilfried Grecksch
Martin-Luther-Universität Halle
Parameter estimate for a linear parabolic fractional SPDE with jumps
A drift parameter estimation problem is studied for a linear parabolic stochas-
tic partial differential equation driven by a multiplicative cylindrical fractional
Brownian motion with Hurst index h ∈]1/2, 1[ and a multiplicative Poisson pro-
cess with values in a Hilbert space. Equations are introduced for the Galerkin
approximations of the mild solution process. A mean square estimation crite-
rion is used for these equations. It is proved that the estimate is unbiased and
weakly consistent for the original problem.

Vitalii Konarovskyi
Universität Leipzig
Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line
The talk is devoted to a model of interacting diffusion particles on the real line. We propose a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction. The construction is based on a new family of measures on the set of real increasing functions as reference measures for naturally associated Dirichlet forms. The processes are infinite dimensional versions of sticky reflecting dynamics on a simplex. We also identify the intrinsic metric leading to a Varadhan formula for the short time asymptotics with the Wasserstein metric for the associated measure valued diffusion. Joint work with Max von Renesse.

Jeanette Köppe
Martin-Luther-Universität Halle
Stochastic optimal control theory for quantum systems
In 1966, E. Nelson established a new interpretation of quantum mechanics, whereby the particles follow some conservative diffusion process, i.e. forward-backward stochastic differential equations (FBSDEs), which are equivalent to the Schrödinger equation (1). Until now, this equivalence has been applied in such a way that a known solution to the Schrödinger equation is used to integrate the stochastic differential equations numerically and analyze the statistical properties of the sample paths. Compared to the options available to treat classical systems this is limited, both in methods and in scope.

However, in analogy to classical mechanics, we show that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton’s principle of least action, allows to derive two aspects (2): i) the Schrödinger equation as the Hamilton-Jacobi-Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton’s equations of motion to the quantum world. We derive their general form for the n-dimensional, non-stationary and the stationary case. The resulting forward-backward stochastic differential equations can be solved numerically without using the solution of the Schrödinger equation, which is done for many different systems, e.g. one- and two-dimensional harmonic oscillator, one-dimensional double-well potential or hydrogen atom.

  1. E. Nelson (1966). Derivation to the Schrödinger Equation from Newtonian Mechanics. Phys. Rev. 150(4), 1079–1085
  2. J. Köppe, W. Grecksch, W. Paul (2017). Derivation and application of quantum Hamilton equations of motion. Ann. Phys. 529(3), 1600251

Carlotta Langer
Martin-Luther-Universität Halle
Iterative Scaling Algorithms and their Applications
Iterative scaling is a widely used method to solve maximum entropy problems.

There are many variant iterative scaling algorithms. In order to reconnect two often-used iterative scaling algorithms, my bachelor thesis categorizes the different types of algorithms and analyses their relations. Four different implemented algorithms are used to study and compare their convergence behaviour.

Björn Schmalfuss
FSU Jena
Local exponential stability for SDE’s driven by an fbm with Hurst parameter in (13,12]

We formulate conditions for existence and uniqueness for SDE’s driven by an fbm with Hurst parameter in (13,12]. We then prove that under particular conditions on the coefficients of this equation for initial conditions in a random neighborhood of zero the solution converges exponentially fast to the trivial solution.

Scott Smith
Stochastic continuity equations with conservative noise
This talk is concerned with regularization by noise for the continuity equation. We will review the classical deterministic theory of Di-Perna/Lions and the regularization by linear, multiplicative noise initiated by Flandoli/Gubinelli/Priola. Finally, I will present a new result, joint with Benjamin Gess, which allows for non-linear, partially degenerate noise in conservative form.

Christoph Trautwein
MPI Magdeburg
Optimal Control of the Stochastic Navier-Stokes Equations
We consider an optimal control problem for incompressible random
flows governed by the stochastic Navier-Stokes equations in a multidi-
mensional domain. The control problem introduced in this talk is mo-
tivated by common strategies, such as tracking a desired velocity field
and minimizing the enstrophy energy. Since existence and uniqueness
results of mild solutions are provided only locally in time, the cost func-
tional has to incorporate stopping times dependent on the controls. We
state an existence and uniqueness result for the optimal control and using
a stochastic maximum principle, we derive necessary optimality condi-
tions. As a consequence, we get an explicit formula of the optimal con-
trol based on the adjoint equation, which is given by a backward SPDE.
Moreover, we show that sufficient optimality conditions are satisfied. This
enables us to obtain unique solutions of control problems constrained by
the stochastic Navier-Stokes equations for two-dimensional as well as for
three-dimensional domains.

Max von Renesse
Universität Leipzig
Couplings, Gradient Estimates and Logarithmic Sobolev Inequality for Langevin Bridges
with Giovanni Conforti
We establish quantitative results about the bridges of the
Langevin dynamics and the associated reciprocal processes. They include an equivalence
between gradient estimates for bridge semigroups and couplings, comparison
principles, bounds of the distance between bridges of different Langevin dynamics, and
a Logarithmic Sobolev inequality for bridge measures.


Date and Location

November 17 - 18, 2017
Max Planck Institute for Mathematics in the Sciences
Inselstraße 22
04103 Leipzig
see travel instructions

Scientific Organizers

Benjamin Gess
MPI für Mathematik in den Naturwissenschaften

Administrative Contact

Katja Heid
MPI for Mathematics in the Sciences
Contact by Email
22.11.2017, 01:27