Rough Analysis and Stochastic Dynamics | Berlin Math Seminars

Rough Analysis and Stochastic Dynamics 📅

Homepage

https://sites.google.com/view/trr388/events/seminars

Institute

Head

Benjamin Gess

Usual time

Thursdays 11am+

Number of talks

33

Comment

The TRR 388 seminars page links to the current Google Spreadsheet for this series; the scraper follows that link automatically.

Thu, 23.07.26 at 11:00

Room MA 748

Jost Pieper Durham

Bridging rough SDEs and rough semimartingales: A martingale fixed point approach

Abstract.

A crucial ingredient in the study of rough SDEs is the notion of a rough stochastic integral. Alongside the stochastic sewing approach to rough SDEs (see [FrizHocquetLê2021] and [AllanPieper2026]), a different notion of rough stochastic integration was introduced in the framework of rough semimartingales in [FrizZorin-Kranich2023]. While the stochastic sewing approach relies on strong integrability assumptions on the driving martingale, the rough semimartingale framework requires only mild integrability assumptions. These reduced requirements allow for an Itô calculus for rough semimartingales under weaker assumptions than those required for the corresponding calculus developed through the stochastic sewing approach, especially without the need for boundedness or growth conditions on the test functions. In this talk, I will discuss the connection between the solution theory for càdlàg rough SDEs via stochastic sewing and the theory of rough semimartingales. I will first discuss the pathwise regularity properties of rough Itô processes by establishing suitable versions of the Kolmogorov continuity criterion in p-variation spaces. Second, I will present a new approach to rough SDEs formulated directly in the space of rough semimartingales through a martingale fixed point argument. As an application, I will discuss a novel rough stochastic Duhamel formula with jumps, which allows one to quickly verify the assumptions of this approach in the setting of semilinear rough SDEs. This is a joint work in progress with Jannis R. Dause and Peter K. Friz.

Thu, 16.07.26 at 11:00

online

Kuan Zhang Shandong U.

Rough BSDEs

Abstract.

We establish an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven by a geometric rough path of finite p-variation, with p < 3. In contrast to earlier approaches based on transformation methods or approximation of smooth drivers, the equation is here, for the first time, solved intrinsically. In particular, this gives meaning to the solution pair (Y, Z) as solution to a hybrid rough stochastic integral equation. The proof relies on an original combination of rough stochastic techniques (via stochastic sewing) and p-variation estimates for martingale transforms (in the spirit of Lépingle estimates), supplemented by Malliavin calculus and John–Nirenberg inequalities for VMO p-variation processes. Under suitable regularity assumption, we prove a strict comparison principle and a priori estimates, followed by existence, uniqueness, and local stability with respect to the terminal condition, the drift coefficient, and the driving rough path. Brownian randomization of the rough path leads connects naturally to BDSDEs. (Joint work with P. Friz, J. Song and H. Zhang)

Thu, 09.07.26 at 11:00

Room MA 748

Paul Gassiat Paris

A path roughness approach to observability in deterministic control systems

Abstract.

We consider deterministic control systems in which only partial observations of the state are available. A classical result of Sussmann (1978) states that, for analytic systems, the class of maximally observing (“universal”) controls is generic among smooth inputs. We revisit this result using path-roughness criteria originating in stochastic analysis and rough path theory. This viewpoint yields direct probabilistic consequences: for instance, Brownian and fractional Brownian sample paths provide universal controls, almost surely. We also show how the same roughness mechanism can be used in output-feedback stabilization, and illustrate this in the case of partially observed bilinear systems. Joint work with Lucas Brivadis (L2S, CentraleSupélec).

Thu, 11.06.26 at 11:00

Room MA 748

Matteo Ravot Licheri Berlin

Towards a Marcinkiewicz Theorem for Expected Signatures with Applications to PDE

Abstract.

In 1939 Marcinkiewicz published a celebrated result which establishes that a random variable with polynomial cumulants is necessarily distributed as a (possibly degenerate) Gaussian. We reformulate this result in the language of signatures and rough paths, and explore the properties of random variables with values in the group like elements of the tensor series algebra. We obtain a partial characterization of such random variables with polynomial cumulants (in the tensor series algebra) as Brownian motions diffused along the free Lie algebra. This endeavor is then connected to positivity-preserving properties of some parabolic PDEs with unbounded coefficients.

Thu, 28.05.26 at 11:00

Room MA 748

Helena Kremp Berlin

Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

Abstract.

When approximating solutions of SPDEs, a basic challenge is that the rate of convergence is limited due to the low time and space regularity of the solution. Considering semilinear SPDEs with additive space-time white noise in space dimension d=1, we introduce a novel numerical scheme which improves the spatial convergence rate from the classical rate 1/2 to rate 3/2. The temporal convergence rate is proven to be 3/4 for our scheme, which enhances the classical rate 1/4 using ideas of previous works (Jentzen, Kloeden ‘08, Jentzen ’11, Djurdjevac, Kremp, Gerencser '24). The talk is based on a work in progress together with Lukas Anzeletti and Mate Gerencser.

Thu, 21.05.26 at 11:00

Room MA 748

Ioannis Gasteratos Berlin

Regularisation by noise for stochastic wave equations

Abstract.

We consider nonlinear Stochastic Wave Equations (SWE), posed on d-dimensional bounded domains and perturbed by additive noise that is white in time, colored in space, with spatial covariance given by a Riesz kernel. Leveraging regularisation-by-noise techniques in infinite dimensions, we show pathwise well-posedness of SWE with Hölder nonlinearities b for d ≤ 4 and (probabilistically) weak existence of solutions for distributional b in all dimensions. Our approach relies on a shifted version of Lê’s stochastic sewing lemma in Hilbert spaces. Time permitting, we shall discuss an application to Smoluchowski-Kramers approximation with irregular coefficients. Ongoing joint work with Oleg Butkovsky (WIAS and HU Berlin) and Michael Salins (Boston University).

Thu, 07.05.26 at 11:00

Room MA 748

Fabio Bugini Berlin

Rough Stochastic Kalman-Bucy Filter

Abstract.

In this talk, we adopt a rough path perspective on the classical Kalman-Bucy filtering problem with conditionally linear dynamics and correlated noise. By treating the observation process as a deterministic rough path, the signal dynamics are formulated as Rough Stochastic Differential Equations (RSDEs) with linear coefficients, in the sense of Friz, Hocquet, and Lê. We show that under a Gaussian initial condition, the distribution of the rough signal process remains Gaussian, and we characterize its mean and covariance as solutions to deterministic Rough Differential Equations (RDEs). In particular, the covariance satisfies a linear-quadratic Riccati RDE, for which we establish global well-posedness and non-explosion results that are of independent interest and, to the best of our knowledge, the first for this class of equations. Finally, we show that, when randomised, the RDEs for the mean and the covariance recover the classical Kalman-Bucy filtering equations. Joint work with P.K. Friz, K. Lê, and H. Zhang (arXiv:2509.11825).

Thu, 30.04.26 at 11:00

Room MA 748

Jannis Dause Berlin

Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities

Abstract.

We introduce a calculus of space–time controlled functions for rough stochastic systems, providing a unified composition rule for evaluating random fields along rough semimartingales. This framework yields a rough stochastic Itô–Wentzell formula under natural and verifiable regularity assumptions. Motivated by recent work of Hude et al. (2024) and, independently, Del Moral and Singh (2022), which established forward–backward perturbation identities for diffusion processes, we show how Itô–Alekseev–Gröbner and backward Itô–Wentzell type representations arise naturally within a rough stochastic framework. This is joint work with Peter K. Friz, Arnulf Jentzen and Jian Song.

Thu, 23.04.26 at 11:00

Room MA 748

Tom Klose Oxford

A Rough Functional Breuer–Major Theorem

Abstract.

We extend the functional Breuer–Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983). This is joint work with Henri Elad Altman (Paris XIII) and Nicolas Perkowski (FU Berlin).

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

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

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

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

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

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 12:00

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, 19.12.24 at 11:00

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 12:00

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. The talk is based on the following manuscripts: F. de Feo, S. Federico, A. Święch, "Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models", SIAM J. Control Optim. 62 (2024), no. 3, 1490–1520. F. de Feo, A. Święch, "Optimal control of stochastic delay differential equations: Optimal feedback controls", arXiv preprint arXiv:2309.05029 (2023). F. de Feo, "Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems", Decis. Econ. Finance (2024) 31 pp. F. de Feo, A. Święch, L. Wessels, "Stochastic optimal control in Hilbert spaces: $C^{1,1}$-regularity of the value function and optimal synthesis via viscosity solutions", arXiv preprint, arXiv:2310.03181 (2023).

Thu, 12.12.24 at 11:00

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 12:00

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, 05.12.24 at 11:00

Harprit Singh University of Vienna, AT

Rough Geometric Integration

Thu, 28.11.24 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, 28.11.24 at 11:00

online

Jost Pieper

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 12:00

Stefanos Theodorakopoulos TU Berlin

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

Abstract.

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, 14.11.24 at 11:00

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)