Rough Analysis and Stochastic Dynamics | Berlin Math Seminars

Rough Analysis and Stochastic Dynamics 📅

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Institute

Usual time

Thursdays 11am+

Number of talks

19

Thu, 29.05.25 at 12:00

on-site

Zhen-Qing Chen U of Washington

Approximation of Liouville Brownian motion

Abstract.

Liouville Brownian motion was introduced as a canonical diffusion process under Liouville quantum gravity. It is constructed as a time change of 2-dimensional Brownian motion by the continuous additive functional associated with a Liouville measure, through a regularizing approximation procedure of the Gaussian free field. In this talk, we are concerned with the question whether one can construct Liouville Brownian motion directly from the Liouville measure. We will present a discrete approximation scheme that in fact works for any time-changed Brownian motion by a Revuz measure that has full quasi support. Based on joint work with Yang Yu.

Thu, 29.05.25 at 12:00

on-site

Gideon Chiusole TU Munich

Towards Abstract Wiener Model Spaces

Abstract.

Abstract Wiener spaces are in many ways the decisive setting for fundamental results on Gaussian measures: large deviations (Schilder), quasi-invariance (Cameron--Martin), differential calculus (Malliavin), support description (Stroock--Varadhan), concentration of measure (Fernique), ... Analogues of these classical results have been derived in the "enhanced" context of Gaussian rough paths and, more recently, regularity structures equipped with Gaussian models. The aim of this talk is to propose a notion of "abstract Wiener model space" that encompasses the aforementioned. More specifically, we focus here on enhanced Schilder type results, Cameron-Martin shifts and Fernique estimates, offering a somewhat unified view on results in Friz-Victoir 2007 and Hairer-Weber 2015.

Thu, 29.05.25 at 12:00

on-site

Filippo De Feo TU Berlin

Optimal control of stochastic delay differential equations via SDEs and PDEs on Hilbert spaces

Abstract.

Optimal control problems involving Markovian stochastic differential equations have been extensively studied in the research literature; however, many real-world applications necessitate the consideration of path-dependent non-Markovian dynamics. In this talk, we consider an optimal control problem of (path-dependent) stochastic differential equations with delays in the state. To use the dynamic programming approach, we regain Markovianity by lifting the problem on a suitable Hilbert space. We characterize the value function $V$ of the problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully non-linear second-order partial differential equation on a Hilbert space with an unbounded operator. Since no regularity results are available for viscosity solutions of these kinds of HJB equations, via a new finite-dimensional reduction procedure that allows us to use the regularity theory for finite-dimensional PDEs, we prove partial $C^{1,\alpha}$-regularity of $V$. When the diffusion is independent of the control, this regularity result allows us to define a candidate optimal feedback control. However, due to the lack of $C^2$-regularity of $V$, we cannot prove a verification theorem using standard techniques based on Ito’s formula. Thus, using a technical double approximation procedure, we construct functions approximating $V$, which are supersolutions of perturbed HJB equations and regular enough to satisfy a non-smooth Ito’s formula. This allows us to prove a verification theorem and construct optimal feedback controls. We discuss how these results extend to the case of delays in the control variable (also) and discuss connections with new results of $C^{1,1}$-regularity of the value function and optimal synthesis for optimal control problems of stochastic differential equations on Hilbert spaces via viscosity solutions.

Thu, 29.05.25 at 12:00

on-site

Stefanos Theodorakopoulos TU Berlin

Topics on mean-field and McKean–Vlasov BSDEs, and the backward propagation of chaos.

Abstract.

We shall present different versions of McKean-Vlasov and mean-field BSDEs of increasing generality, and the notion of backward propagation of chaos. We will then discuss some of the technical difficulties associated with the corresponding limit theorems and see some of their immediate corollaries and rates of convergence. Finally, we will introduce the concept of stability with respect to data sets for the backward propagation of chaos, and state the intermediate results that allowed us to prove its validity under a natural framework.

Thu, 29.05.25 at 12:00

online

Huilin Zhang

Stochastic control and filtering via stochastic rough paths

Abstract.

In this talk, I would like to tell the story about applications of recently developed theory of rough SDE theory ([FHL'21]). More precisely, we solve two major problems: 1. pathwise stochastic control problem; 2. robust stochastic filtering. For the first problem, we introduce a new interpretation to this "ill-posed" problem via rough SDEs, and then introduce the well-known Dynamical Programming Principle and Pontryagin's Maximum Principle to this problem. For the second problem, we build the robust filtering by rough SDEs, and moreover, we provide an approximation to the optimal filter by a discretised rough SDE with an optimal convergence rate. This talk is based on joint works with P. Friz, K. Le and U. Horst.

Thu, 20.02.25 at 11:00

Helena Kremp

Thu, 06.02.25 at 11:00

on-site

Sebastian Grube Bielefeld

Strong solutions to degenerate SDEs and uniqueness for degenerate Fokker-Planck equations

Abstract.

In this talk we will present a method for obtaining probabilistic strong solutions to large classes of SDEs with unbounded and discontinuous drift and diffusion coefficients, where the latter are allowed to degenerate. We will discuss a general approach based on the superposition principle and a ‚restricted‘ Yamada-Watanabe theorem. Furthermore, we will show uniqueness results for the corresponding linear Fokker—Planck equations, building upon the technique of Röckner and Zhang (CRM, 2010).

Thu, 30.01.25 at 12:00

on-site

Jannis Dause HU

Rough Stochastic Optimal Control Problems and Duality

Abstract.

Classical stochastic optimal control is heavily based on the Markovianity of the underlying noise e.g. through the use of the dynamic programming principle and subsequently HJB-equations. Motivated by the increasing interest in optimal control of rough and non-Markovian systems (e.g. driven by fBm), we present a novel duality approach to tackle such problems. Inspired by the approach of Rogers [SICON, '07] in discrete time and the continuous-time, rough extension of Diehl-Friz-Gassiat [APPL MATH OPT, '17], we introduce a penalty term for the pathwise control problem based on a (functional) Taylor expansion and prove an approximative duality result by suitable tightness properties of solutions to controlled RDEs. We further discuss the viscosity theory needed to analyze the rough, pathwise problem.

Thu, 30.01.25 at 11:00

on-site

Lorenzo Dello Schiavo Rome TV

Massive Particle Systems, Wasserstein Brownian Motions, and the Dean--Kawasaki SPDE

Abstract.

Let W be a conservative, ergodic Markov diffusion on some arbitrary state space M, converging exponentially fast to equilibrium. We consider: (1) Systems of up to countably many massive particles in M, with finite total mass. Each particle is subject to an independent instance of the noise W, with volatility the inverse mass carried by the particle. We prove that the corresponding infinite system of SDEs has a unique solution, for every starting configuration and every distribution of the masses in the infinite simplex. (2) Solutions to the Dean--Kawasaki SPDE with singular drift, driven by the generator L of W. We prove that the equation may be given rigorous meaning on M, and that it has a unique `distributional’ solution. This extends Konarovskyi--Lehmann--von Renesse's `ill-posedness vs. triviality' to the case of infinitely many massive particles. (3) Diffusions with values in the space P of all probability measures on M, driven by the geometry induced by L. (4) In the case when M is a manifold, differential-geometric and metric-measure Brownian motions on P induced by the geometry of optimal transportation and reversible for a normalized completely random measure. We show that all these objects coincide.

Thu, 23.01.25 at 11:00

on-site

Sebastian Kassing TUB

Training Dynamics of Artificial Neural Networks in Supervised Learning

Abstract.

In machine learning, loss functions are typically non-convex and contain multiple minima. Though this poses a substantial hurdle in the theoretical analysis, stochastic optimization methods, such as stochastic gradient descent and stochastic heavy ball, perform surprisingly well in practice. In this talk we discuss a few recent results concerning the theoretical analysis of probabilistic programs used for training neural networks combining approaches from stochastic analysis, geometry and stochastic optimization. We derive fast convergence rates for Polyak’s heavy ball on non-convex objective functions and approximate the dynamics of Riemannian stochastic gradient descent by a diffusion process on the manifold of the search space.

Thu, 16.01.25 at 11:00

on-site

Ioannis Gasteratos TUB

Rough differential equations for volatility

Abstract.

We introduce a canonical way of performing the joint lift of a Brownian motion W and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser, and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where $\mathbf{X}$ is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with W) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data. This is joint work with Ofelia Bonesini (LSE), Emilio Ferrucci (Oxford) and Antoine Jacquier (Imperial College London).

Thu, 09.01.25 at 12:00

online

Peter Friz TUB

Rough stochastic differential equations

Abstract.

The core of my talk is devoted to explain rough stochastic differential equations (RSDEs), a common generalization of Ito SDEs and Lyons RDEs. With concrete motivation from (I) non-linear filtering theory, (II) pathwise stochastic control and (III) a recent rough PDE approach to pricing in non-Markovian stochastic volatility models, I will then indicate all the progress made possible with RSDEs.

Thu, 19.12.24 at 11:00

on-site

Benjamin Robinson Klagenfurt

A regularized Kellerer theorem in arbitrary dimension

Abstract.

We present a multidimensional extension of Kellerer's theorem on the existence of mimicking Markov martingales for peacocks, a term derived from the French for stochastic processes increasing in convex order. For a continuous-time peacock in arbitrary dimension, after Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale Itô diffusion. Moreover, we provide counterexamples to show, in dimension at least 2, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale. Key ingredients in our existence proof are so-called Bass martingales from martingale optimal transport, a mimicking theorem for Itô processes, and a novel compactness result for martingale diffusions. This is joint work with Gudmund Pammer (ETH Zürich) and Walter Schachermayer (Universität Wien). To appear in Annals of Applied Probability.

Thu, 12.12.24 at 11:00

on-site

Leonid Berlyand Penn State University

Enhancing Accuracy in Deep Learning using Marchenko-Pastur Distribution

Abstract.

We begin with a short overview of Random Matrix Theory (RMT), focusing on the Marchenko-Pastur (MP) spectral approach. Next, we present recent analytical and numerical results on accelerating the training of Deep Neural Networks (DNNs) via MP-based pruning ([1]). Furthermore, we show that combining this pruning with L2 regularization allows one to drastically decrease randomness in the weight layers and, hence, simplify the loss landscape. Moreover, we show that the DNN’s weights become deterministic at any local minima of the loss function. Finally, we discuss our most recent results (in progress) on the generalization of the MP law to the input-output Jacobian matrix of the DNN. Here, our focus is on the existence of fixed points. The numerical examples are done for several types of DNNs: fully connected, CNNs and ViTs. These works are done jointly with PSU PhD students M. Kiyashko, Y. Shmalo, L. Zelong and with E. Afanasiev and V. Slavin (Kharkiv, Ukraine).

Thu, 05.12.24 at 11:00

on-site

Harprit Singh University of Vienna, AT

Rough Geometric Integration

Thu, 28.11.24 at 11:00

online

Jost Pieper and Andrew Allan

An application of RSDEs to robust filtering with jumps

Abstract.

Finding a robust representation of the conditional distribution of a signal given a noisy observation is a classical problem in stochastic filtering. When the signal and observation are correlated through their noise, Crisan, Diehl, Friz, and Oberhauser (2013) demonstrated that such a robust representation cannot generally exist as a functional on the space of continuous functions but instead can be found as a function on the space of geometric rough paths. In this talk, I will discuss my ongoing work with Andrew Allan and Josef Teichmann, on the application of the relatively new theory of rough stochastic differential equations (RSDEs) to stochastic filtering problems of correlated jump diffusions, with the aim to establish a robust representation of the filter in this setting.

Thu, 14.11.24 at 11:00

on-site

Marco Rehmeier TU Berlin

p-Brownian motion and the p-Laplacian

Abstract.

We construct a stochastic process, more precisely, a (nonlinear) Markov process, which is related to the parabolic p-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) Laplacian.

Thu, 24.10.24 at 11:00

online

Masato Hoshino Osaka

Regularity-integrability structure and its application

Abstract.

I will report on a series of studies: [1] arXiv:2310.07396, [2] arXiv:2310.10202 (joint work with I. Bailleul), and [3] arXiv:2408.04322 (joint work with R. Takano). In [1], we provide elementary proofs for the key analytic theorems in the theory of regularity structures: the reconstruction theorem and the multilevel Schauder estimate, using the operator semigroup approach. Another aim of [1] is to introduce the new framework of "regularity-integrability structures (RIS)", which might be suitable for situations involving both regularity and integrability exponents, such as Besov spaces. In [2], as an application of RIS, we provide a short proof of the probabilistic convergence theorem for a class of random models including BPHZ models. In [3], we extend the semigroup approach from [1] to singular modelled distributions.

Thu, 17.10.24 at 11:00

Goncalo Dos Reis Edinburg

Ito-Wentzell-Lions formula for measure dependent random fields under full and conditional measure flows

Abstract.

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functionals random field of Itô type depending on measures. We distinguish the full- and marginal-measure flow cases. Derivatives with respect to the measure components are understood in the sense of Lions. This talk is based on joint work with V. Platonov (U. of Edinburgh)