Conference on Stochastic Analysis and Stochastic Partial Differential Equations

A theory of stochastic geometric mechanics

Large deviations for Brownian bridges with terminal densities

Stabilization of Dynamical Systems by Noise with Jumps

Computable least-squares estimator for drift in linear SPDEs with additive fractional noise 

Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations 

Optimal regularity in time and space for stochastic porous medium equations

Least-squares estimator for the drift parameter in the linear SPDEs with additive noise

Stochastic integration with respect to canonical α-stable cylindrical Lévy processes

Stochastic Predictability and Applications on Stochastic Differential Equations

On fractional conservation laws with noise

 Controllability of some semilinear parabolic SPDEs

Fractional integral equations with weighted Takagi-Landsberg functions

On maximal and global solutions for stochastic shallow water models

In this talk I will present the analytical properties of a stochastic shallow water model which has been derived using the Location Uncertainty (Mémin, 2014) approach. We prove that there exists a unique maximal strong solution and a unique global weak solution, using a method which relies on a Cauchy approximating sequence. Our method is very convenient as although the strong solution exists in a higher order Sobolev space, we show that it suffices to prove the Cauchy property in L^2. This is different from the approaches currently used in the literature which rely mainly on tightness arguments.

A linear stochastic biharmonic heat equation: hitting probabilities

We find hitting probabilities estimations for the solution of a linear stochastic biharmonic biharmonic heat equation driven by white noise. We first study the anisotropies of the canonical metric of the solution, in particular when the spatial dimension d=2, they include a z(-log(z))^(1/2) term. Applying the criteria proved in previous works, we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets. This yield results on the polarity of sets and on the Hausdorff dimension of the path process. 

This is joint work with M. Sanz-Solé. 

Utility maximisation and change of variable formulas for time-changed dynamics 

In this paper we derive novel change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time change is an increasing stochastic process with finitely many jumps in a bounded set of the positive half-line and is independent of the Brownian motion. As an application we consider the problem of maximising the expected utility of the terminal wealth in a semimartingale setting, where the semimartin-gale is written in terms of a time-changed Brownian motion and a finite variation process. To solve this problem, we use an initial enlargement of filtration and our change of variable formulas to shift the problem to a maximisation problem under the enlarged filtration for models driven by a Brownian motion and a finite variation process. The latter problem can be solved by using martingale properties. Then applying again the change of variable formula, we derive the optimal strategy for the original problem for a power utility and for a logarithmic utility. This is based on a joint work with Giulia Di Nunno, Hannes Haferkorn, and Michèle Van-
maele.

Mean reflected stochastic differential equations with two constraints

Adrián Falkowski | Nicolaus Copernicus University in Toru

The stochastic heat equation with multiplicative Lévy noise: Existence, intermittency, and multifractality

Carsten Chong | Columbia University

In this talk, we consider the stochastic heat equation driven by a multiplicative Léevy noise in arbitrary dimension d >= 1. We prove the existence of solutions under an optimal condition if d = 1; 2 and a close-to-optimal condition if d >= 3. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. Furthermore, for any p > 0, we derive upper and lower bounds on the moment Lyapunovexponents of order p of the solution. One of our most striking findings is that the solution to the SHE is strongly intermittent for any non-trivial Lévy measure, at any disorder intensity, in any dimension d >= 1. The absence of a phase transition from a non-intermittent to an intermittent regime in d >= 3 is in sharp contrast with the SHE on the lattice or the SHE with Gaussian noise. If time permits, I will further discuss the multifractal behavior of intermittency islands in snapshots of the solution. This talk is based on joint works with Quentin Berger, Hubert Lacoin and Péter Kevei.

An affine infinite-dimensional stochastic volatility model

Sonja Cox|  University of Amsterdam

I will first briefly explain what an affine stochastic process is. Such processes have received a considerable amount of attention in the past years due to their tractability and (relative) flexibility For example, in 2011 Cuchiero, Filipovic, Mayerhofer, and Teichmann provided a characterization of all affine processes taking values in the cone of non-negative semi-definite matrices. The motivation for doing so was to construct tractable finite-dimenstional stochastic volatility models.
Our goal was to obtain similar infinite-dimensional stochastic volatility models. To this end, we established existence of affine processes in the cone of positive self-adjoint Hilbert-Schmidt operators. In my talk, I will discuss the details of our result, the challenges of the infinite-dimensional setting, and some open questions.

Self-Avoiding Models of Moving Polymers and Surfaces

Carl Mueller  |  University of Rochester (This is joint work with Eyal Neuman)

Polymer models give rise to some of the most challenging problems inprobability and statistical physics. We typically model a polymer using a random walk, where the time parameter n of the walk represents distance along the polymer starting from one end. That is, we imagine that the polymer is built up by adding molecules one by one at random angles. We usually include a self-avoidance term, re ecting the idea that different parts of the polymer cannot be in the same place at the same time. A diffcult problem, unsolved in the most important physical cases, is to predict the end-to-end distance or radius of the polymer.

In this talk, I will discuss two extensions of the random polymer model.

1. Moving polymers can be modeled by stochastic partial differential equations. If the polymer takes values in one-dimensional Euclidean space, we give fairly sharp upper and lower bounds for its radius. We find that there is more stretching than in typical one-dimensional polymer models that do not have time dependence.
2. Random surfaces can be modeled by elastic manifolds, also called discrete Gaussian free fields. These models originate in quantum field theory. If the dimensions of the parameter space and the range are the same, we can derive bounds on the radius of the polymer. These bounds are fairly sharp in two dimensions.

We will explain the models mentioned above and give an outline of our proof techniques.

Transport noise models from two-scale systems with additive noise in fluid dynamics

Arnaud Debussche  |  École Normale Supérieure de Rennes

Taming Uncertainty and Profiting from Randomness

Michael Röckner  |  Universität Bielefeld

The title of this talk is a short form of the title of a major research project (a socalled Collaborative Research Centre, funded by the German Research Foundation, DFG), currently running at Bielefeld University. It is based on the claim that randomness, which is ubiquitous and cannot be avoided, neither in the natural sciences, nor in the humanities, has in fact two “faces“, one is a kind of a nuisance, which destroys one’s observations or measurements, is hard to quantify and causes uncertainty about the correctness of one’s final conclusions. So, one has to look for means and to develop tools for “taming“ it. On the other hand, in many cases randomness can help to understand observed phenomena which otherwise would remain a mystery, and in addition its mathematical analysis may lead to a deeper understanding of the underlying mechanisms, thus leading to resolving seemingly contradictory conclusions, which with no doubt is a valuable “profit“. In the talk I shall present some examples for the above claim and try to explain what mathematics, in particular stochastic analysis, offers for both “taming uncertainty“ and “profiting from randomness“. One eminent instance for both the latter will be the mathematical approachto understanding dynamical processes under the influence of “noise“, which appear in all sciences. The technical term of this mathematical approach is “stochastic partial differential equations“, but despite this for non-mathematicians maybe a bit “terrifying“ name, I think these can be understood quite well on an intuitive level from a certain view point, as I shall try to explain in the talk.

Optimal control of Volterra type equations: from finite to infinite dimensions and return

Giulia di Nunno  |  University of Oslo

We consider a stochastic optimal control problem for Volterra type forward dynamics, which are typically non-Markovian in nature. A direct application of dynamic programming is not feasible. So we propose to lift the problem to infinite dimensions, where we can recover Markovianity. For this we work with Banach-valued processes.  In this framework we tackle the optimal control problem by studying the corresponding Hamilton-Jacobi-Bellman equation, which can be formulated thanks to the appropriate lift. Finally, we reconnect the solution of the infinite dimensional problem to the original finite dimensional one.

Rough stochastic Analysis

Peter Friz  | TU and WIAS Berlin

We build a hybrid theory which seamlessly combines classical stochastic – and rough analytic considerations. In particular, we give, for the first time, direct meaning to rough Ito diffusions driven simultaneously by a Brownian motion B (in Ito sense) and a rough path W (in Lyons’ sense). This can be viewed as partial robustification of SDEs with two Brownian noise components, typical in filtering situations. In presence of some Markovian structure, we are led to classes of rough/stochastic partial differential equations, including non-linear ones of HJB or mean-field type. Compared with previous approaches based on flow-transformation this approach offers an intrinsic understanding of the underlying process, well-adapted to discrete approximations and with minimal regularity demands. Propagation of chaos for particle systems with common noise is another application of our theory.

Joint work with K. Le (TU Berlin) and A. Hocquet (TU Berlin).

Waving Marta Goodbye

Samy Tindel  |  Purdue University

In this talk I will first give a brief overview of some standard stochastic PDE models. Then I will describe some of Marta’s fundamental contributions to the theory of stochastic wave equations. This includes challenging problems such as the definition of noisy wave equations in higher dimensions, as well as further properties of the solution. Those results are based on nontrivial extensions of Itô type integrals and advanced Malliavin calculus tools, for which I intend to give a short and non technical introduction. Eventually I will give an account on more recent pathwise approaches to noisy wave equations, which are still the object of active research.