BEGIN:VCALENDAR PRODID:-//eluceo/ical//2.0/EN VERSION:2.0 CALSCALE:GREGORIAN BEGIN:VEVENT UID:c19602ddc4e34639897bc6c9276440b3 DTSTAMP:20260514T140950Z SUMMARY:2024 Conference on Modern Topics in Stochastic Analysis and Applica tions (in honour of Terry Lyons’ 70th birthday) DESCRIPTION:Stochastic Analysis is a branch of mathematics dealing with the analysis of dynamical systems evolving under the influence of randomness. In the last 30 years\, it developed into one of the most active areas of mathematics worldwide. UK-based mathematicians have pioneered its recent d evelopments. A wide range of powerful and novel stochastic tools have been developed in recent years\, and with impressive diverse applications with in and beyond mathematics\, including physics\, chemistry\, biology\, comp uter science and social sciences. \nPerhaps the most influential contribut ion in this area has been Lyons’ theory of rough paths\, a mathematical language to describe the effects a stream can generate when interacting wi th non-linear dynamical systems. It has had a fundamental impact on exten ding Ito’s theory of SDEs beyond semimartingales and on the development of Hairer’s Fields medal winning work on regularity structures\, which provides a robust solution theory for many singular stochastic PDEs arisi ng from physics. In recent years\, rough path theory has started to play a key role in the design of state-of-the-art machine learning algorithms fo r processing noisy and high-dimensional data streams in a wide range of c ontexts including finance\, cybersecurity\, precision-medicine\, and bioen gineering.  \nDuring this time\, the Stochastic Analysis Group at Imperi al College London has grown to be one of the strongest and most active in Europe and internationally. The group proposes to organize a five-day conf erence to gather experts from different areas of theoretical and applied s tochastic analysis to investigate and discuss current and future direction s of this field\, with special emphasis on the interplay between theory an d applications. A special session is planned with presentations from indus trial and academic practitioners. This conference will mark the 70th birth day of Prof. Terry Lyons.\nOver the course of the conference\, participant s will have the opportunity to present their latest research\, exchange id eas and collaborate with peers. Through plenary talks\, panel discussions\ , and interactive sessions\, the conference will provide a platform for pa rticipants to learn from one another and gain new insights into the latest trends and techniques in stochastic analysis.\nPOSTER\nOrganisers\nThe co nference is jointly organised with Thomas Cass (Imperial College London)\, Ilya Chevyrev (University of Edinburgh)\, Dan Crisan (Imperial College Lo ndon)\, James Foster (University of Bath)\, Christian Litterer (University of York)\, Hao Ni (University College London) and Cristopher Salvi (Imper ial College London).\nFunders\nLondon Mathematical Society\, DataSig (EPSR C Project Grant EP/S026347/1)\, Making Cubature on Wiener Space Work (EPSR C Project Grant EP/V005413/1)\, Imperial College London (Cecilia Tanner Fu nd).\nInvited speakers: titles and abstracts\n \n\n\n\n\nPeter Friz (TU B erlin)\nTitle: Rough analysis of rough volatility models\nAbstract: The qu estion what rough paths have to do with finance has received many answers in recent years\, this talk is devoted to some these. I will start by repo rting on recent work that identifies the weak rate for a class of rough (B ergomi type) volatility models. In a second part\, I will present a rough PDE based extension of the Romani-Touzi formula\, describing the law of s ome asset under partial conditioning\, applicable in particular to local r ough stochastic volatility models. Finally\, I will present an extension o f Gatheral’s diamond calculus\, motivated by rough Heston in forward var iance form\, to expected signatures\, offering systematic computations in general semimartingale models. Credit to numerous coworkers will be given in the talk.\n\n\n\nRené Carmona (Princeton University)\nTitle: Control o f Conditional Processes\n\n\n\nSandy Davie (University of Edinburgh)\nTitl e: Central limit bounds in Vaserstein metrics\nAbstract: The Vaserstein me trics from optimal transport theory seem to be natural measures of distanc e between probability measures\, and several authors have used them to giv e bounds for normal approximations as in the Central Limit Theorem. In par ticular Rio in 2011 gave quite sharp estimates for such bounds. I will des cribe how (under some moment conditions) asymptotic expansions can be obta ined\, in negative powers of the sample size\, for  Vaserstein distances between the distribution of a sample mean and the corresponding normal dis tribution. These expansions throw some light on  a question raised by Vil lani.\n\n\n\nFelix Otto (Max Planck Institute\, Leipzig)\nTitle: Regularit y structures: more geometry and less combinatorics\nAbstract: Singular sto chastic partial differential equations are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. T he guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow Hairer’s regularity str uctures\, which however we re-interpret as providing an informal\, and ev entually rigorous\, parameterization of the infinite-dimensional nonlinea r solution manifold. We systematically follow this more geometric/analytic than combinatorial point-of-view: Instead of appealing to an expansion in dexed by trees\, we consider all partial derivatives w.r.t. the “constit utive” function defining the nonlinearity. Instead of a Gaussian calculu s guided by Feynman diagrams arising from pairing nodes of two trees\, we consider derivatives w.r.t. the noise\, i.e. Malliavin derivatives. We int erpret the Malliavin derivative of the parameterization as an approximate tangent vector to the solution manifold\, which yields a sparse representa tion in terms of the parameterization itself\, and paves the way for its s tochastic estimate. Ultimately\, this gives a characterization of the solu tion manifold that is oblivious to the divergent counter terms. This is wo rk with L. Broux\, R. Steele\, P. Linares\, M. Tempelmayr\, and P. Tsatsou lis\, based on work with J. Sauer\, S. Smith\, and H. Weber.\n\n\n\nMartin Hairer (EPFL & Imperial College London)\nTitle: What happens at H = 1/4?\ nAbstract: It has been known since the work of Coutin & Qian that fraction al Brownian motion has a canonical rough path lift for H > 1/4\, while onl y non-canonical lifts are known when this condition fails. We show that th e suitably rescaled lift of a mollified fractional Brownian motion converg es to a limiting “pure area” rough path with entries given by standard Brownian motions that are asymptotically independent of the underlying fr actional Brownian motion.\n\n\n\nMichael Rőckner (University of Bielefeld )\nTitle: Non-linear Fokker-Planck-Kolmogorov equations as gradient flows on the space of probability measures\nAbstract: We propose a general metho d to identify nonlinear Fokker–Planck–Kolmogorov equations (FPK equati ons) as gradient flows on the space of Borel probability measures on Rd wi th a natural differential geometry. Our notion of gradient flow does not d epend on any underlying metric structure such as the Wasserstein distance\ , but is derived from purely differential geometric principles. Moreover\, we explicitly identify the associated energy functions and show that thes e are Lyapunov functions for the FPK solutions. Our main result covers cla ssical and generalized porous media equations\, where the latter have a ge neralized diffusivity function and a nonlinear transport-type first-order perturbation.\n\n\n\nYves Le Jan (Université Paris-Saclay)\nTitle: Moran model with selection\nAbstract: We consider a biparental model in which th e population is initially divided in two groups of different fitness. We d etermine the time evolution of the contribution of each initial group to t he genome of the total population.\n\n\n\nDavid Nualart (University of Kan sas)\nTitle: Gaussian fluctuations for spatial averages of the stochastic heat equation\nAbstract: The purpose of this talk is to survey several res ults on quantitative central limit theorems for spatial averages of the so lution to the stochastic heat equation driven by a Gaussian noise with spa tial  homogeneous covariance. The estimates are based on the combination of Stein’s method for normal approximations and the techniques of Mallia vin calculus.  We will also discuss the case of the parabolic Anderson mo del driven by a Gaussian noise colored in time\, where the corresponding m ild equation is formulated using the Skorohod integral.\n\n\n\nOfer Zeitou ni (Weizmann Institute of Science)\nTitle: Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices\nAbstract: I will descri be the determination to leading order of the maximum of the characteristi c polynomial for Wigner matrices and $\\beta$-ensembles. In the special c ase of Gaussian-divisible Wigner matrices\, the method provides universal ity of the maximum up to tightness. These are the first universal results on the Fyodorov–Hiary–Keating conjectures for these models\, and in p articular it answers the question of optimal rigidity for the spectrum of Wigner matrices. The proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlat ed fields and Gaussian multiplicative chaos. Joint work with Paul Bourgade and Patrick Lopato\n\n\n\nGerard Ben Arous (Courant Institute\, NY Univer sity)\nTitle: Dynamical spectral transition for optimization in very high dimensions\nAbstract: The dynamics of optimization\, needed for Machine Le arning tasks\, take place in a very high dimensional setting. But it seems that in fact the most important aspects really happen in a much smaller d imension. In recent work with Reza Gheissari (Northwestern)\, Aukosh Jagan nath (Waterloo) we gave a general context for the existence of projected “effective dynamics” of SGD in very high dimensions for “summary sta tistics” in much smaller dimensions. These effective dynamics (and\, in particular\, their so-called ‘critical regime”) define a dynamical sys tem in finite dimensions which may be quite complex\, and rules the perfor mance of the learning algorithm. The next step is to understand how the sy stem finds these “summary statistics”.  This is done in the last wor k with the same authors and with Jiaoyang Huang (Wharton\, U-Penn). This i s based on a dynamical spectral transition (the BBP transition of Random M atrix Theory): along the trajectory of the optimization path\, the Gram ma trix or the Hessian matrix develop outliers whose eigen-spaces carry these effective dynamics. I will illustrate the use of this point of view on a few central examples of ML:  classification for Gaussian mixtures\, and t he XOR task.\n\n\n\nBen Hambly (University of Oxford)\nTitle: Particle sys tems with feedback and networks of neurons\nAbstract: Interacting particle systems with feedback have been investigated recently as they arise in mo dels in finance and neuroscience as well as having connections to Stefan p roblems. I will discuss the features of these models and focus on the case of the leaky integrate and fire model in neuroscience. In this setting we will discuss the scaling limit of the empirical measure of the particle s ystem\, showing the existence and uniqueness of solutions for an SPDE whic h describes the evolution of the whole system. We will also discuss the di fferent effects that can result from adjusting the feedback parameter.\n\n \n\nAlison Etheridge (University of Oxford)\nTitle: Forwards and backwards in spatially heterogeneous populations\nAbstract: We introduce a broad cl ass of individual based models that might describe how spatially heterogen eous populations live\, die\, and reproduce. Our primary interest is in un derstanding how genetic ancestry spreads across geography when looking bac k through time in these populations. A novelty is that by explicitly split ting reproduction into two phases (production of juveniles and their matur ation) we produce a framework that not only captures models which when sui tably scaled converge to classical reaction diffusion equations\, but also ones with nonlinear diffusion that exhibit quite different behaviour. Thi s is joint work with Tom Kurtz (Madison)\, Peter Ralph (Oregon) and Ian Le tter and Terence Tsui (Oxford).\n\n\n\nChristian Bayer (WIAS Berlin)\nTitl e: Signatures for stochastic optimal control\nAbstract: Models with memory play an increasingly important role in many applications\, from finance t o molecular dynamics. In a stochastic setting\, memory means that the unde rlying stochastic process is not a Markov process. Such processes are part icularly challenging for stochastic optimal control\, as most state-of-the -art methods for solving stochastic optimal controls problems heavily rely on the Markov property. Building on earlier works by Terry Lyons\, we sho w that paths signatures allow us to efficiently solve several classes of s tochastic optimal control problems even when the underlying state process is not a Markov process\, We provide theoretical analysis and numerical ap plications\, with special emphasis on optimal stopping and optimal executi on problems.\n\n\n\nZhongmin Qian (University of Oxford)\nTitle:  On the duality of conditional laws of diffusions and applications\nAbstract: In t his talk I will report the duality of conditional diffusion processes obta ined recently for a large class of time in-homogeneous diffusion processes . By using the duality of conditional laws new type of Feynman-Kac formula s are established for solutions of some parabolic equations. Some applicat ions to random vortex system and to compressible flows will be discussed.\ n\n\nAndrew Stuart (Caltech)\nTitle: Gradient Flows for Sampling: Mean-Fie ld Models\, Gaussian Approximations and Affine Invariance\nAbstract: Sampl ing a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task m ay be cast as an optimization problem over all probability measures\, by c hoice of a suitable energy function. Then an initial distribution can be e volved to the desired minimizer (the target distribution) via a gradient f low with respect to a chosen metric. The choice of the energy and the metr ic lead to different approaches and it is of interest to understand their role. We provide theoretical insights into these choices. Having chosen an energy and a metric\, development of an actionable algorithm requires app roximation of the gradient flow. Mean-field models\, whose law is governed by the gradient flow in the space of probability measures\, may be identi fied\; particle approximations of these mean-field models form the basis o f algorithms. The gradient flow approach is\nalso the basis of algorithms for variational inference\, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians or Gau ssian mixtures\; the underlying gradient flow is restricted to the paramet erized family. Numerical results are presented to illustrate the resulting methodologies. Joint work with Y. CHEN (NYU)\, D.Z. HUANG (PKU)\, J. HUAN G (U Penn) and S. REICH (Potsdam).\n\n\nJosef Teichman (ETH)\nTitle: On t he relation of real analytic functions on path spaces and signature expans ions.\nAbstract: Signature expansions and associated kernel techniques are important tools in approximation theory on path spaces. We show some rela tions to more classical notions of real analytic functions on path spaces and to some concepts of invariant theory. We then head towards a better un derstanding of reproducing kernel Hilbert spaces related to signature kern els. (joint works with Christa Cuchiero\, Walter\nSchachermayer\, and Vale ntin Tissot-Daguette).\n\n\nHans Buehler (XTX Markets)\nTitle: Learning to Trade\nAbstract: We discuss advances in applying reinforcement learning t o risk managing a variety of financial problems.\n\n\nSalvador Ortiz-Lator re (University of Oslo)\nTitle: Deep learning methods for the stochastic f iltering problem\nAbstract: The stochastic filtering equations govern the evolution of the conditional distribution of a signal process given partia l\, and possibly noisy\, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applicatio ns\, including numerical weather prediction\, finance and engineering.  I n this talk we will present an approach based on the combination of the PD E method (also known as the splitting-up method) for solving the stochasti c filtering problem and a deep learning method to represent the solution o f the PDE involved. Our approach follows a recent stream of research into deep learning based approximations of high dimensional PDEs and related wo rks within the context of stochastic optimal control. Joint work with Dan Crisan (Imperial) and Alexander Lobbe (Imperial).\n\n\n\nHarald Oberhauser (University of Oxford)\nTitle: Random Surfaces and Higher Dimensional Alg ebra\nAbstract: The path signature provides a structured description of an (unparametrized) path by representing it as an element in a non-commutati ve group\; path concatenation turns into group multiplication\, and path-r eversal into group inversion. We study a generalization to surfaces\, that is maps indexed by a two-dimensional domain. It turns out that so-called double groups (aka crossed modules) are rich enough to faithfully capture the compositional structure of (unparametrized) surfaces and that signatur e objects arise as the solution of differential equation. This has many co nsequences\, one of them is that analogous to how the expected path signat ure can characterize laws of stochastic processes indexed by “time” on e can characterize laws of random surfaces in an algebraic structured and principled way\, thus providing a natural “characteristic function” fo r laws of random surfaces. We develop this construction up to a Young regi me. Joint work with Darrick Lee.\n\n\n\nPaola Arrubarrena (Imperial Colleg e London)\nTitle: Anomaly Detection on Radio Astronomy Data using Signatur es\nAbstract: An anomaly detection methodology is presented that identifie s if a given observation is unusual by deviating from a corpus of non-cont aminated observations. The signature transform is applied to the streamed data as a vectorization to obtain a faithful representation in a fixed-dim ensional feature space. This talk is applied to radio astronomy data to id entify very faint radio frequency interference (RFI) contaminating the res t of the data.\n\n\n\nMaud Lemercier (University of Oxford)\nTitle: A High Order Solver for Signature Kernels\nAbstract: Signature kernels are at th e core of several machine learning algorithms for analysing multivariate t ime series. The kernel of two bounded variation paths (such as piecewise l inear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However\, this approach becomes considera bly less practical for highly oscillatory input paths\, as they have to be resolved at a fine enough scale to accurately recover their signature ker nel\, resulting in significant time and memory complexities. To mitigate t his issue\, we first show that the signature kernel of a broader class of paths\, known as smooth rough paths\, also satisfies a PDE\, albeit in the form of a system of coupled equations. We then use this result to introdu ce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric -rough paths) can b e approximated by piecewise smooth rough paths\, one can replace the PDE w ith rapidly varying coefficients in the original Goursat problem by an exp licit system of coupled equations with piecewise constant coefficients der ived from the first few iterated integrals of the original input paths. Wh ile this approach requires solving more equations\, they do not require lo oking back at the complex and fine structure of the initial paths\, which significantly reduces the computational complexity associated with the ana lysis of highly oscillatory time series.\n\n\n\nAdeline Fermanian (Califra is)\nTitle: Dynamic Survival Analysis with Controlled Latent States\nAbstr act: We consider the task of learning individual specific intensities of c ounting processes from a set of static variables and irregularly sampled t ime series. We introduce a novel modelization approach in which the intens ity is the solution to a controlled differential equation. We first design a neural estimator by building on neural controlled differential equation s. In a second time\, we show that our model can be linearized in the sign ature space under sufficient regularity conditions\, yielding a signature- based estimator which we call CoxSig. We provide theoretical learning guar antees for both estimators\, before showcasing the performance of our mode ls on a vast array of simulated and real-world datasets from finance\, pre dictive maintenance and food supply chain management\n\n\n\nHoratio Boedha rdjo (University of Warwick)\nTitle: Lack of Gaussian Tail for integrals a long fractional Brownian motions\nAbstract: A result of Kusuoka and Strooc k states that the solutions to elliptic stochastic differential solutions\ , where vector fields have uniformly bounded derivatives of all orders\, h ave Gaussian tails. For rough differential equations driven by fractional Brownian motions with Hurst parameter H and with same assumptions on vecto r fields as before\, a celebrated result of Cass-Littterer-Lyons establish ed a max(1+2H\,1/2)-Weibull upper bound for the tail probability of the so lution. Establishing a general lower bound seems difficult but we will dis cuss the particular case of integral along fractional Brownian motions of smooth functions with uniformly bounded and non degenerate derivatives of all orders. Joint work with Xi Geng. \n\n\n\nSyoiti Ninomiya (Tokyo Insti tute of Technology)\nTitle: New deep learning machine architecture based o n higher-order weak approximation algorithms for SDEs\nAbstract: A new dee p-learning neural network architecture based on high-order weak approximat ion algorithms for stochastic differential equations is proposed. The arch itecture enables deep learning machines to learn martingales efficiently.  The behavior of deep neural networks based on this architecture when app lied to the financial derivatives pricing problem is also reported. The cr ux of this new architecture lies in the higher-order weak approximation al gorithms for SDEs of Runge-Kutta type\, in which the approximation is real ised by iterative substitutions and their linear combinations.\n\n\n\nStef ano Goria (Thymia)\nTitle: Signatures for Machine Learning-Driven Mental H ealth Biomarkers\nAbstract: Mental health care lags far behind physical he alth in many dimensions\, with the lack of accessible and accurate biomark ers for its symptoms being one of the causes. In recent years\, we have be en witnessing an increased interest in mental health biomarkers particular ly based on machine learning techniques applied to extremely rich data sou rces\, such as speech or video. We review here our recent results from app lying path signatures to speech and video data in order to predict the cog nitive test scores used in diagnosing ASD and ADHD\, positioning this meth od more broadly as a powerful tool to support the development of machine l earning-driven biomarkers.\n\n\n\nBlanka Horvath (University of Oxford)\nT itle: Pathwise methods for pricing\, trading and market generation\nAbstra ct: The emergence of machine learning solutions for pricing\, hedging and trading has opened up new horizons for addressing these tasks under a larg e variety of models and market conditions. The emergence of these solution s has also provided the first applications where more nuanced properties o f markets became necessary than what classical models could provide. In th is setting\, generative models and pathwise methods rooted in rough paths have proven to be powerful from several perspectives. At the same time\, a ny model – a traditional stochastic model or a market generator – is a t best an approximation of market reality\, prone to model misspecificatio n and estimation errors. The unfading question\, “how to furnish a model ling setup with tools that can address the risk of the discrepancy between model and market reality” also gains new colour as multitude of new pos sibilities arise under the pathwise perspective.\n\n\n\n \nSchedule\n \n \n\n\n\n\nMonday 22\nTuesday 23\nWednesday 24\nThursday 25\nFriday 26\n\n\ n8.30-8.55\nRegistration\n\n\n\n\n\n\n9 – 9.45\nChristian Bayer\nAdeline Fermanian\nSandy Davie\nOfer Zeitouni\nJosef Teichmann\n\n\n9.45 – 10.3 0\nRené Carmona\nPeter Friz\nMichael Rőckner\nYves Le Jan\nSyoiti Ninomi ya\n\n\nCoffe break (10.30 – 11)\n\n\n\n\n\n\n\n11 – 11.45\nMaud Lemer cier\nMartin Hairer\nAlison Etheridge\nDavid Nualart\nHans Buehler\n\n\nLu nch break (11.45 – 14)\n\n\n\n\n\n\n\n14 – 14.45\nBlanka Horvath\nBen Hambly\nPhD session\nGerard Ben Arous\nSalvador Ortiz-Latorre\n\n\n14.45 – 15.30\nHoratio Boedhardjo\nFelix Otto\nPhD session\nZhongmin Qian\nAnd rew Stuart\n\n\nCoffe break (15.30 – 16)\n\n\n\n\nClosing remarks\n\n\n1 6 – 16.45\nStefano Goria\nHarald Oberhauser\nPhD session\nPaola Arruebar rena\n\n\n\n\n\n\n\n\n\n\n\n\n\nConference dinner (invitation only)\nPhD s tudents dinner (invitation only)\n\n\n\n\n\n URL:https://www.imperial.ac.uk/events/168741/conference-on-modern-topics-in -stochastic-analysis-and-applications-in-honour-of-terry-lyons-70th-birthd ay/ DTSTART;TZID=Europe/London:20240422T090000 DTEND;TZID=Europe/London:20240426T170000 LOCATION:340\, Huxley Building\, South Kensington Campus\, Imperial College London\, London\, SW7 2AZ\, United Kingdom END:VEVENT BEGIN:VTIMEZONE TZID:Europe/London BEGIN:DAYLIGHT DTSTART:20240422T090000 TZNAME:BST TZOFFSETTO:+0100 TZOFFSETFROM:+0100 END:DAYLIGHT END:VTIMEZONE END:VCALENDAR