Abstracts

Dr. Christian Bayer - WIAS Berlin

Title: Learning rough volatility

Abstract: Rough stochastic volatility is a new modelling framework for equity prices in
mathematical finance, which leads to great fits to market data with only very
few free parameters. Rough volatility processes, however, fail to be
semi-martingales and, even worse, the Markov property does not
hold. This means that many conventional numerical methods for option
pricing, model
calibration, optimal control and other tasks fail altogether or become much
more costly to apply. We show how machine learning techniques can be used to
calibrate rough volatility models to market prices, and discuss how to use
signatures in the context of optimal control of rough volatility models.

 

Cristopher Salvi - Oxford University 

Title: Areas-of-areas on Hall trees shuffle generate the shuffle algebra

Abstract: We consider the coordinate iterated integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct  area(.,.) which is directly related to the concept of signed-area. We consider Hall sets and we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.

 

Dr. Harald Oberhauser - Oxford University

Title: Gaussian Processes indexed by (rough) paths

Abstract: The space of (geometric or branched rough) paths can be equipped with a natural inner product using the signature lift. This so-called signature kernel can be used as a covariance function to define a real-valued Gaussian process indexed by paths. Besides being an interesting mathematical object, it connects ideas from stochastic analysis with Bayesian inference. As a concrete application, we develop a variational approach to learn the posterior of this process indexed by paths.

 

Rosa Preiss - Technische Universität Berlin

Title: Algebraic methods for Signatures of Paths: Hopf, Zinbiel and Tortkar

Abstract: It is well known that the full signature of a path can be represented as a linear functional on the shuffle algebra, which is a Hopf algebra when equipped with the shuffle product and the deconcatenation coproduct. Both product and coproduct respectively appear in a relation for signatures of paths, the shuffle identity on the one hand and Chen's identity on the other hand. The shuffle algebra can also be seen as a Zinbiel algebra, and there is another relation for signatures corresponding to the non-associative non-commutative halfshuffle product. By forming the commutator of the halfshuffle product, we get a third algebraic structure, a Tortkara algebra, which so far isn't very well understood for dimension (i.e. number of algebra generators) greater than two. But there is again a simple statement that relates the Tortkara algebra coming from the shuffle algebra with computing areas of areas of the underlying path of a given signature.

 

Patrick Kidger - Oxford University

Title: Deep Signature Transforms

Abstract: The signature transform has previously been treated as a fixed feature transformation, on top of which a model may be built. The talk will decribe a general approach which combines the advantages of the signature transform with modern deep learning frameworks. In particular, by learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation.

 

Imanol Perez -Oxford University

Title: Optimal execution with rough path signatures

Abstract: We consider a well-studied optimal execution problem under little assumptions on the underlying price process. We do so by using signatures from rough path theory, that allows converting the original problem into a more computationally tractable problem. We include a few numerical experiments where we show that our methodology is able to retrieve the theoretical optimal execution speed for several problems studied in the literature, as well as some cases not included in the literatture. We also study some estensions of our framework to other settings.

 

Dr. Peter Friz - Technische Universität Berlin 

Title: Rough Transport, Revisited

Abstract: We revisit the work of Diel et al (2017) concerning an intrinsic theory

of rough partial differential equations. In contrast to previous works, also by Bailleul--Gubinelli, we treat the case of general roughness.

 

Dr. Josef Teichmann - ETH Zürich

Title: Representing dynamics through random dynamical systems

Abstract: We re-discover the paradigm of reservoir computing in
stochastic or rough differential equations and prove generalization
bounds. This opens a new perspective on randomness in recurrent neural
networks and on the approximation of stochastic or rough differential
equations. Applications to time series prediction and term structure
problems are discussed.

 

Dr. Marianne Clausell - Universitè de Lorraine

Title: The signature method in machine learning. Applications to textual data analysis and time series clustering

Abstract:  Multidimensional time series analysis is a fundamental problem in many applications as neurosciences, finance, textual data analysis....The signature method has recently gained attention in the machine learning community due to its nice computational properties and interpretable nature.

 

Dr. Torstein K. Nilssen - University of Agder

Title: Rough perturbations of the Navier-Stokes equation

Abstract: In this talk I will present a rough path perturbation of the Navier-Stokes system which preserves circulation. The presentation will focus on how to define an intrinsic notion of solution of the equation, as well as a discussion on well-posedness using the vorticity formulation . 

 

Patric Bonnier - Oxford University

Title:  Signature Cumulants and independence of Stochastic Processes

Abstract: "The sequence of so-called Signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates the study of so-called Signature cumulants. To do so, an elementary combinatorial approach is developed and used to show that in the same way that cumulants relate to the lattice of partitions, Signature cumulants relate to the lattice of so-called "ordered partitions". This is used to give a new characterisation of independence of multivariate stochastic processes."

 

Dr. Joscha Diehl - Greifswald Univeristy

Title:  Iterated sums and applications

Abstract:  A proven tool for extracting information from a discrete time-series
is to interpolate it linearly and calculate the iterated-integrals
signature. It turns out though that for a one-dimensional time-series
the extracted information consists only in the total increment over
the whole time interval. Using discrete integration a lot more data is
gathered, while at the same time preserving some of the invariant
properties that the classical signature enjoys. I discuss the
algebraic structures involved and applications. This is joint work
with Kurusch Ebrahimi-Fard (NTNU) and Nikolas Tapia (WIAS Berlin).

 

 

Published Oct. 28, 2019 8:48 AM - Last modified Nov. 13, 2019 3:16 PM