Titles + Abstracts 

Speaker: Christian Bayer

Title: Path signature methods for pricing of Bermudan options 

Abstract: Path signatures provide provide a powerful tool for solving stochastic optimal control problems in non-Markovian models, in particular enabling the pricing of American options under rough volatility. In this talk, we discuss classical primal and dual approaches extended to rough volatility models with the help of the signature transform, as well as a further approach based on the signature kernel.  We demonstrate and compare the performance within the popular rough

Heston and rough Bergomi models. Joint work with Luca Pelizzari and Jia-Jia Zhu. 

Speaker: Thomas Cass

Title: Signature kernels as universal limits of randomised controlled differential equations 

Abstract: We review recent results on one-dimensional signature kernels and demonstrate how tools from random matrix theory and free probability can be employed to realise these kernels as universal limits of randomised controlled differential equations (CDEs). Of particular interest is the class of CDEs describing the Cartan development of paths onto matrix Lie groups. For the general linear group GL(n), the ordinary signature kernel arises in the limit. For the unitary group U(n), the limiting kernel is characterised as the solution to a novel quadratic functional equation. The structure of this equation is determined by the Schwinger–Dyson (SD) equations associated with the joint moments of a collection of free semicircular random variables. We explore properties of this kernel and propose strategies for its numerical solution. This work is based on joint research with Will Turner.

Speaker: Ilya Chevyrev

Title: 2D signatures and sewing lemma

Abstract: In this talk, I will describe a recent proposal for the definition of a 2D (surface) signature based on crossed modules. I will especially highlight a two-dimensional and non-commutative version of the sewing lemma which provides an “intrinsic” construction of the surface signature. Based on joint work with Joscha Diehl, Kurusch Ebrahimi-Fard, and Nikolas Tapia.

Speaker: Christa Cuchiero

Title: Conditional polynomial McKean-Vlasov SDEs

Abstract: We study a new class of McKean-Vlasov stochastic differential equations (SDEs), possibly with common noise, applying the theory of time-inhomogeneous polynomial processes. The drift and volatility coefficients of these SDEs depend on the state variables themselves as well as their conditional moments in a way that mimics the standard polynomial structure. Our approach leads to new results on the existence and uniqueness of solutions to such conditional McKean-Vlasov SDEs which were, to the best of our knowledge, not obtainable using standard methods. Indeed, the special form of the characteristics causes the conditional moments to become an autonomous standard Ito-SDE driven by the common Brownian noise. Provided that a pathwise unique solution exists to this SDE, we can establish an existence, uniqueness and tractability theory that covers a large class of conditional McKean Vlasov SDEs beyond the standard conditions of Lipschitz continuity and uniform ellipticity. Replacing the common Brownian noise by a rough (deterministic) one, we also discuss variants of such McKean-Vlasov equations in the realm of rough stochastic differential equations considered by Friz et. al.  Another important extension can be obtained by including the (conditional) expectation of the process' signature so that the equation does not only depend on the marginals laws but rather on the law on path space. The polynomial approach can also be adapted to an affine structure where the drift and volatility may depend more generally on the characteristic function of the process’ marginals and thus on the full law and not only on the moments.

The talk is based on joint work with Janka Möller.

Speaker: Joscha Diehl

Title: Tensor-to-Tensor Models with Fast Iterated Sum Features

Abstract: Data in the form of images or higher-order tensors is ubiquitous in modern deep learning applications. Owing to their inherent high dimensionality, the need for subquadratic layers processing such data is even more pressing than for sequence data.

We propose a novel tensor-to-tensor layer with linear cost in the input size, utilizing the mathematical gadget of ``corner trees'' from the field of permutation counting. In particular, for order-two tensors, we provide an image-to-image layer that can be plugged into image processing pipelines. On the one hand, our method can be seen as a higher-order generalization of state-space models. On the other hand, it is based on a multiparameter generalization of the signature of iterated integrals (or sums). 

Joint work with Rasheed Ibraheem, Leonard Schmitz, Yue Wu.

Speaker: Nina Drobac

Title: Signed, Sealed, Predicted: Time Series Forecasting with Signatures

Abstract: Recent challenges in electricity load forecasting, such as the COVID-19 pandemic and the energy crisis, have highlighted the need for adaptive predictive models. In this talk, we propose a time series forecasting framework that involves fitting a penalised linear regression model to the signature transformation of the covariate series.

We begin by briefly presenting the signature and motivating its use as a non-parametric feature set for sequential data, which effectively captures non-linear relationships and past dependencies. We then present theoretical guarantees, establishing universality and stationarity properties of signature features when computed over sliding windows. To bridge the gap between theory and practice, we describe a sequential algorithm that leverages the algebraic structure of the signature space for more efficient computation. The presentation concludes with experimental results on both simulated and real data, demonstrating the framework’s potential for electricity load forecasting.

Speaker: Xin Guo

Title: TBA 

Abstract: TBA

Speaker: Paul Hager

Title: Well-Posedness and Stability of Signature Controlled Differential Equations

Abstract: Among various approaches to lifting path-dependent dynamics to infinite-dimensional state dynamics, using the Signature is particularly natural, as it arises directly from the free development of the system itself and robustly captures a wide range of roughness. A commonly accepted piece of folklore holds that 'within reason’ any path-dependent system can be modeled by a suitable signature differential equation.

Our theoretical results substantiate this folklore. In particular, we provide a local Lipschitz and linear growth condition for well-posedness, along with a stability result for universal approximation. We also address some of the topological subtleties entailed by comparing the path-dependent and infinite-dimensional perspectives. Maintaining the lifted system allows us to consider several spin-offs of signature differential equations - such as mean-reverting and finite memory signatures - which naturally embed into the context of smooth rough paths. The approximation results are illustrated by some numerical simulations. 

Based on joint work with Tomás Carrondo, Christa Cuchiero, and Fabian Harang.

Speaker: Ni Hao 

Title: Sig-DEG for Distillation: Making Diffusion Models Faster and Lighter 

Abstract: Diffusion models have achieved state-of-the-art results in generative modelling but remain computationally intensive at inference time, often requiring thousands of discretization steps. To this end, we propose Sig-DEG (Signature-based Differential Equation Generator), a novel generator for distilling pre-trained diffusion models, which can universally approximate the backward diffusion process at a coarse temporal resolution. Inspired by high-order approximations of stochastic differential equations (SDEs), Sig-DEG leverages partial signature to efficiently summarize Brownian motion over sub-intervals and adopts a recurrent structure to enable accurate global approximation of the SDE solution. Distillation is formulated as a supervised learning task, where Sig-DEG is trained to match the outputs of a fine-resolution diffusion model on a coarse time grid. During inference, Sig-DEG enables fast generation, as the partial signature terms can be simulated exactly without requiring fine-grained Brownian paths. Experiments demonstrate that Sig-DEG achieves competitive generation quality while reducing the number of inference steps by an order of magnitude. Our results highlight the effectiveness of signature-based approximations for efficient generative modeling.

Speaker: Blanke Hovrath 

Title: TBA 

Abstract: TBA

Speaker: Eduardo Abi Jaber 

Title: Chasing Stationarity: The Fading-Memory Signature 

Abstract: We introduce the fading-memory (FM) signature, a time-invariant transformation of an infinite path that serves as a mean reverting analogue of the classical path signature. The FM-signature retains many of the key properties of path signatures, including a suitably modified Chen’s identity and a linearization property. Unlike the standard signature, it acts as a “stationarized” representation, making it a compelling feature for time-series analysis and signal processing. For the FM-signature of time-extended Brownian motion, we establish stationarity, exponential ergodicity in the Wasserstein distance, and derive an explicit formula à la Fawcett for its expected value. Based on joint work with Dimitri Sotnikov

Speaker: Darrick Lee

Title: Thin Homotopy and the Signature of Piecewise Linear Surfaces

Abstract: In this talk, we introduce an algebraic construction for piecewise linear paths and surfaces, which leads to a natural definition of the signature for such objects. Furthermore, we discuss the notion of thin homotopy of surfaces, which involves non-local behaviour (as opposed to the case of paths), and discuss the injectivity of the surface signature with respect to thin homotopy classes of piecewise linear surfaces. Based on joint work with Francis Bischoff.

Speaker: Maud Lemercier

Title: Log-PDE Methods for Rough Signature Kernels

Abstract: Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. In this talk, I will extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. This system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation to rough signature kernels. The talk is based on joint work with Cristopher Salvi and Terry Lyons.

Speaker: Luca Pelizzari

Title: On the Volterra signature

Abstract: In this talk, we introduce the Volterra signature, which naturally arises in the study of (rough) Volterra equations. These equations play a central role in modeling random systems with memory, and we show that the Volterra signature serves as a practical feature map in such settings. We support this by presenting several algebraic and analytic properties of Volterra signatures, including an efficient representation for piecewise linear paths, invariance, universality, and an explicit expressions for moments. This is based on an ongoing project with P. Hager, F. Harang and S. Tindel.

Speaker: Francesca Primavera

Title:  Functional Itô-formula and Taylor expansions for non-anticipative maps of càdlàg rough paths  

Abstract: We derive a functional Itô-formula for non-anticipative maps of rough paths, based on the approximation properties of the signature of càdlàg rough paths. This result is a functional extension of the Itô-formula for càdlàg rough paths (by Friz and Zhang (2018)), which coincides with the change of variable formula formulated by Dupire (2009) whenever the functionals' representations, the notions of regularity, and the integration concepts can be matched. Unlike these previous works, we treat the vertical (jump) perturbation via the Marcus transformation, which allows for incorporating path functionals where the second order vertical derivatives do not commute, as is the case for typical signature functionals. As a byproduct, we show that sufficiently regular non-anticipative maps admit a functional Taylor expansion in terms of the path's signature, leading to an important generalization of the recent results by Dupire and Tissot-Daguette (2022). 

Speaker: Cristopher Salvi

Title: Expressivity and Parallelism in Structured State Space Models  

Abstract:  I will present two frameworks advancing the theory and implementation of sequence models. SLiCEs (Structured Linear Controlled Differential Equations) generalize linear controlled differential equations with input-dependent structured transition matrices—block-diagonal, sparse, or Walsh–Hadamard—that retain the expressivity of dense models while enabling parallel-in-time computation. ParallelFlow reinterprets linear attention via matrix-valued state space models and flow discretization, connecting chunking mechanisms to rough path theory. These tools unify and extend existing architectures such as DeltaNet, yielding efficient models with provable expressivity and improved runtime. Based on joint works with Ben Walker, Lingy Yang, Nicola M. Cirone and Terry Lyons. 

Speaker: Nikolas Tapia

Title: Stability of Deep Neural Networks via discrete rough paths

Abstract: Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input

data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer, S. Breneis and P. K. Friz. 

Speaker: Samy Tindel

Title: Some rough paths and signatures techniques in reinforcement learning 

Abstract: In this talk I will start by reviewing some classical results relating machine learning problems with control theory. I will mainly discuss some very basic notions of supervised learning as well as reinforcement learning. Then I will show how noisy environments lead to very natural equations involving rough paths. This will include a couple of motivating examples. In a second part of the talk I will try to explain the techniques used to solve reinforcement learning problems with a minimal amount of technicality. In particular, I will focus on rough HJB type equations and their respective viscosity solutions. Eventually, I will discuss the implementation of q-learning techniques, which make use of signature expansions. 

This talk is based on a joint work with Estepan Ashkarian (Purdue, Math), Prakash Chakraborty (Penn State) and Harsha Honnappa (Purdue, Industrial Engineering).

Speaker: William Turner

Title: Numerical methods for scaling limits of randomised unitary path developments.  

Abstract: Recent work has used scaling limits of randomised path developments to construct kernels on unparameterised path space. For instance, the scaling limit of GL_N​ path developments recovers the signature kernel (Muca Cirone et al.), while randomised U_N​ developments yields a contraction of signature terms against monomials of freely independent semicircular random variables. The resulting kernel satisfies a quadratic functional equation derived via the Schwinger–Dyson equations. A direct discretisation of this equation leads to a numerical scheme with cubic complexity in the path length.

In this work, we propose two alternative numerical approaches. First, inspired by the work of Lemercier and Lyons on the ordinary signature kernel, we study path developments driven by 2-smooth rough paths and derive the corresponding functional equation. Second, we introduce a randomised sparse path development using the Pauli basis of the unitary Lie algebra. This structure enables the design of a quantum algorithm that efficiently estimates the trace of the path development in high dimensions. This is joint work with Tom Cass, Sam Crew, Cris Salvi and Jack Jacquier.