# 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 L ^{q} spaces and duality principle with forward stochastic Volterra integral equations **

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

E = L^{q}(S,Σ,μ), 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***MPI MIS***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 |u_{t}|^{q−2}u_{t} and a source term of the type |u|^{p−2}u. 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.

- E. Nelson (1966). Derivation to the Schrödinger Equation from Newtonian Mechanics. Phys. Rev. 150(4), 1079–1085
- 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 (1∕3,1∕2] **

We formulate conditions for existence and uniqueness for SDE’s driven by an fbm with Hurst parameter in (1∕3,1∕2]. 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***MPI MIS***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

Germany

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