PhD Seminar: Advanced Stochastic Modelling

Upcoming Seminars in SS 2026 

 

  • Speaker 1: Qi Feng
  • Title: Branched Signature Models
  • Time: Tuesday, May 19, 13:15 – 14:15
  • Abstract: In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [Hairer-Kelly. Annales de l'Institue Henri Poincar\'e, Probabilit\'es et Statistiques 51, no. 1 (2015)], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $\Psi$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications. We demonstrate the effectiveness of the proposed framework through two representative applications: rough volatility calibration and optimal execution.
  • Speaker 2: Enrico Maria Ferrari
  • Title: Learning Topological Multiparameter Filtrations: An Application to Medical Image Classification
  • Time: Tuesday, May 19, 14:15 – 15:00
  • Abstract: This talk provides an introduction to Topological Data Analysis (TDA), focusing on persistent homology and its recent extension to multiparameter persistent homology. After presenting the main ideas and mathematical foundations underlying these tools, applications to data analysis will be discussed. Particular emphasis will be placed on the interaction between TDA and Deep Learning, where topological methods can be incorporated directly into the optimization process to enhance model interpretability. A medical image classification problem will be presented as a motivating case study.
  • Location: Seminarraum 15 (OG01), Kolingasse 14
  • Speaker: Guido Gazzani
  • Title: Ultra-Short-Term Volatility Surface (joint with F. M. Bandi, N. Fusari and R. Renò)
  • Time: Tuesday, April 28, 13:15 – 14:45
  • Location: Seminarraum 15 (OG01), Kolingasse 14
  • Abstract: Options with maturities below one week, hereafter ultra-short-term options, have seen a sharp increase in trading activity in recent years. Yet, these instruments are difficult to price jointly using classical pricing models due to the pronounced oscillations observed in the at-the-money implied-volatility term structure across ultra-short-term tenors. We propose Edgeworth++, a parsimonious jump–diffusion model featuring a nonparametric stochastic volatility component, which provides flexibility in capturing the implied-volatility smiles for each tenor, combined with a deterministic shift extension, which allows the model to fit rich at-the-money implied-volatility term structures across tenors. We derive a local (in tenor) expansion of the process characteristic function suited to value ultra-short-term options. The expansion leads to fast and accurate option pricing in closed form via standard Fourier inversion. We discuss the benefits of the proposed approach relative to natural benchmarks.
  • Speaker: Tomás Carrondo
  • Title: Stochastic Optimization Algorithms and Rough Invariance Principles
  • Time: Tuesday, May 5, 13:15 – 14:45
  • Location: Seminarraum 15 (OG01), Kolingasse 14
  • Abstract: Stochastic Gradient Descent (SGD) algorithms underpin the training of many learning models. To better understand these optimization methods, many works have introduced continuous-time surrogates, ranging from the classical gradient flow ODE to more recent diffusion models known as stochastic modified equations (SMEs). Crucially, however, these surrogates apply to SGD with replacement, rather than to the widely used SGD without replacement (SGDo), where training proceeds by exhausting the dataset before beginning a new epoch. To help address this gap, we recast rough recursions, a well-established framework in rough path theory, as a tool for deriving continuous-time limits of stochastic optimization algorithms. Rough recursions allow for arbitrary noise sources, without imposing a specific probabilistic structure, and provide a principled passage from discrete algorithms to continuous-time diffusions. This passage is justified by a rough invariance principle: a Donsker-type result in rough path topology. With the goal of applying this machinery to Random Reshuffling SGD (RR-SGD), a particular variant of SGDo, we prove a rough invariance principle for exchangeable increments. Specifically, we show that random walks obtained by randomly permuting a fixed sample converge in rough path topology to an enhanced Brownian bridge. We then apply this result to RR-SGD in the case of quadratic loss functions and discuss future directions, including a control problem for determining optimal learning schedules for RR-SGD. This talk is based on joint work with Luca Pelizzari.
  • Speaker: Nikolas Tapia 
  • Title: Orthogonal Polynomials on Path-Space
  • Time: Tuesday, May 12, 13:15 – 14:45
  • Location: Seminarraum 15 (OG01), Kolingasse 14
  • Abstract: We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in L^p functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an L^2-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.
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