Christian Bayer

Optimal stopping with signatures

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature X<∞ associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically.The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. of financial or electricity markets. (Based on joint work with Paul Hager, Sebastian Riedel, and John Schoenmakers)

Zdzislaw Brzezniak

Wave maps with values in the sphere defined on the future light cone with random data on the boundary.

We study wave maps with values in S^d, defined on the future light cone {|x| \leq t} \subset R^{1+1}, with prescribed data at the boundary {|x| = t}. Based on the work of Keel and Tao, we prove that the problem is well-posed for locally absolutely continuous boundary data. We design a discrete version of the problem and prove that for every absolutely continuous boundary data, the sequence of solutions of the discretised problem converges to the corresponding continuous wave map as the mesh size tends to 0. Next, we consider the boundary data given by the S^d-valued Brownian motion. We prove that the sequence of solutions of the discretised problems has an accumulation point for the topology of locally uniform convergence. We argue that the resulting random field can be interpreted as the wave-map evolution corresponding to the initial data given by the Gibbs distribution.

This talk is based on a joint work with Jacek Jendrej.

Dan Crisan

Particle filters for geophysical fluid dynamics models

Particle filters are recursive algorithms designed for the numerical inference of partially observed dynamical systems. Even the most basic particle filter (the so-called SIR algorithm or the bootstrap particle filter) can perform well when the observed dynamical system evolves in small to moderate dimensional spaces. In high dimensions, the bootstrap particle filter. Nevertheless, in recent years, a plethora of new methodologies have been introduced to alleviate this apparent curse of dimensionality.

In this talk, I will present a particle filter that incorporates three additional add-on procedures: nudging, tempering and jittering. The particle filter is tested on a two-layer quasi-geostrophic model with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed.

Joscha Diehl

Multiparameter (iterated) sums

Iterated sums (or integrals) have proven very beneficial in time series analysis. I demonstrate how ideas from this one-parameter setting can be used to study multiparameter data, for example, images. I willalso sketch the Hopf algebraic background and the relation to iterated integrals. This is joint work with Leonard Schmitz (University of Greifswald).

Goncalo Dos Reis

High order splitting methods for stochastic differential equations

In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver.

Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain“iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology.

James Foster

Markov Chain Cubature for Bayesian Inference

Markov Chain Monte Carlo (MCMC) is widely regarded as the “go-to” approach for computing integrals with respect to posterior distributions in Bayesian inference. Whilst there are a large variety of MCMC methods, many prominent algorithms can be viewed as approximations of stochastic differential equations (SDEs). For example, the unadjusted Langevin algorithm (ULA) is obtained as an Euler discretization of the Langevin diffusion and has seen particular interest due to its scalability and connections to the optimization literature. In this talk, we will discuss an alternative to Monte Carlo for SDE simulation known as “Cubature on Wiener Space” (Lyons and Victoir, 2004). In the cubature paradigm, SDE solutions are represented as acloud of particles and propagated via deterministic formulae. However, such formulae can dramatically increase the number of particles, and thus SDE cubature requires “distribution compression” to be practical. We show that by applying cubature to ULA and resampling particles in a spatially balanced manner, we can obtain a novel gradient-based particle method for Bayesian inference. Finally, we will conclude by comparing our algorithm to MCMC and Stein Variational Gradient Descent on Gaussian mixture and Bayesian logistic regression models.

Francesco Galuppi

Signatures from the geometric viewpoints

Signature tensors are powerful tools that appears not only in stochastics and non-commutative algebra,but has applications in financial mathematics, data analysis and machine learning. I will take yet another point of view and introduce signature tensors form the algebraic geometry viewpoint. Indeed, they naturally define several families of varieties. Studying the geometry of these varieties leads to a new perspective on signature tensors.

Satoshi Hayakawa

Positively weighted kernel quadrature and a refined analysis of Nyström approximation

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples. Furthermore, we will talk about a refined error analysis of Nyström approximation and a consequent theoretical guarantee of the proposed kernel quadrature, which improves the results already published in NeurIPS 2022. This is joint work with Harald Oberhauser and Terry Lyons.

Christian Litterer

A brief overview of cubature on Wiener space and the computational constructions of its measures

We give a brief overview of the cubature on Wiener space approach developed by Lyons and Victoir and a closely related algorithm due to Kusuoka. In the second half of the talk we will explain some of the methods and challenges encountered in the constructions of high degree cubature formulas on Wiener space. We explain how recombination can be applied in this context and adapted to produce more efficient cubature measures.

Terry Lyons

We will survey the growing contribution of rough path based ideas, such as the signature of a path, in understanding real world streamed data.

A more comprehensive account can be found at www.DataSig.ac.uk/papers

Andrew Donald Mcleod

Generalised Recombination Interpolation Method (GRIM)

Finding sparse approximation of functions allows one to reduce computational complexity. Sparse representations have been applied in a wide range of areas including compressed sensing, image processing, facial recognition, and DNA denoising. In joint work with Terry Lyons, we introduce the Generalised Recombination Interpolation Method (GRIM) for finding sparse approximations of functions that are initially given as a linear combination of some (large) number of simpler functions. GRIM is a hybrid of dynamic growth-based interpolation techniques and thinning-based reduction techniques. In this talk we will introduce GRIM, provide example problems to which it may be applied, illustrate the known theoretical guarantees for the output of GRIM, and empirically demonstrate its good performance on problems involving machine learning datasets.

Syoiti Ninomiya

Patch dividing algorithms for high-order recombination and its application to weak approximations of stochastic differential equations

Those algorithms to perform the patch dividing which are necessary when weakly approximating SDEs using the recombination method developed in [Lyons and Litterer, 2012] are proposed. Refining Lyons-Litterer’s error estimation provides the patch radius criteria on which these algorithms are based. Numerical examples of the calculation of option prices under the Heston model are also presented.

Joint work with Yuji Shinozaki.

Harald Oberhauser

Capturing Graphs with Hypo-elliptic Diffusion

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator called the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In particular, this gives a structured approach to capturing long-range dependencies on graphs that is robust to pooling. Joint work with Csaba Toth, Darrick Lee, and Celia Hacker.

Salvador Ortiz-Latorre

Pathwise approximations for the solution of the non-linear filtering problem

In this talk we will first briefly introduce a class of high order approximations for the solution of the stochastic filtering problem. Then, we will derive their robust representation in the spirit of Clark and Davis. In particular, we will show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. This is a joint work with Dan Crisan and Alexander Lobbe.

Raúl Penaguião

The discrete signature Veronese variety

Diehl, Ebbrahimi-Fard and Tapia introduced the iterated-sums signature of a time series, a discrete version of a path signature that has time warping properties. In this talk we will discuss about the discrete signature variety, the points traced out by this signature. We will compute its dimension and follow the study done on the continuous case by Amendola, Sturmfels and Friz. We will however see that even for small parameter values the variety is already quite complex and high-dimensional.

Alvise Sommariva

On some theorems by Tchakaloff, Davis and Wilhelmsen and their applications

In this talk, we start introducing the well-known Tchakaloff theorem on a multivariate compact domain Ω⊂ Rd. Then we show how, starting from cubature rules on Ω with positive weights and interior nodes (i.e. of PI-type), whose algebraic degree of precision ADE is equal to L and the number of nodes higher than the dimension η of the polynomial space P_L(Ω) of total degree L, we can extract rules of PI-type but with at most η nodes, by means of Lawson-Hanson algorithm. After some numerical examples, we proceed recalling some results byDavis and Wilhelmsen and show how they imply that we can determine rules of PI-type, with ADE equal to L, if the moments w.r.t. to a basis of P_L(Ω) are available as well as if we can evaluate numerically the characteristic function X_Ω. We conclude proposing examples of this novel meshless approach on bivariate domains whose boundary can be tracked by piecewise NURBS and on polyhedra.