The IMPRS Combo IV is an interdisciplinary and introductory workshop for PhD students and PostDocs in their early career. It aims to bring together all IMPRS students and encourage them to learn about a wider spectrum of mathematics.

The theme of this workshop will be the Path Signature, an infinite collection of tensors whose entries are iterated integrals of the path’s coordinate functions. Originally introduced by Kuo-Tsai Chen in the 1950s within the framework of differential geometry and algebraic topology, path signatures have since become a cornerstone of rough path theory in stochastic analysis. Its applications range from quantitative finance to machine learning and mathematical physics.

This workshop will bring together different perspectives on this versatile object and its uses. We will explore the algebraic and analytic structures underlying path signatures, with a view towards its applications in finance and data science.

Confirmed Speakers:

• Guido Gazzani (University of Verona)
• Darrick Lee (University of Edinburgh)
• Rosa Preiß (TU Berlin)
• Gabriel Tarziu (LMU Munich)

Abstracts for mini-courses:

Guidio Gazzani - Signatures: From Theory to Financial Applications

In this course, we shall focus on the use of the signature as a universal linear regression basis for continuous functionals of paths in financial applications. After a brief introduction to continuous rough paths, we show how to embed continuous semimartingales into the rough path setting. Our main focus is on the signature of semimartingales, one of the key modeling tools in finance.

In the financial applications we have in mind, a crucial quantity to compute is the expected signature of a given underlying process. Surprisingly, this can be achieved for broad classes of diffusions using techniques from affine and polynomial processes. More precisely, we show how the signature process of these diffusions can be embedded within the framework of affine and polynomial processes. This not only provides a straightforward method for computing the expected signature of a polynomial diffusion but also gives access to its Fourier–Laplace transform via related Riccati-type ODEs.

Regarding financial applications, we discuss signature-based asset price models to address the long-debated SPX/VIX joint calibration problem. We consider a stochastic volatility model in which the volatility dynamics are given by a linear function of the (time-extended) signature of a primary process, assumed to be a polynomial diffusion. We obtain closed-form expressions for the squared VIX, exploiting the fact that the truncated signature of a polynomial diffusion is itself a polynomial diffusion.

By augmenting such a primary process with the Brownian motion driving the stock price, we can express both the log-price and the squared VIX as linear functions of the signature of the corresponding augmented process. This property can be efficiently used for pricing and calibration purposes: since signature samples can be precomputed, the calibration task can be split into offline sampling and a standard optimization. We also propose a Fourier pricing approach for both SPX and VIX options, exploiting the fact that the signature of the augmented primary process is an infinite-dimensional affine process. For both SPX and VIX options, we obtain highly accurate calibration results, demonstrating that this model class can solve the joint calibration problem without requiring jumps or rough volatility.

This course is based on joint work with Christa Cuchiero, Janka Moeller, and Sara Svaluto-Ferro.

Rosa Preiß - Shuffles, tensors and varieties: Algebra of path signatures

The iterated-integrals signature comes with a rich algebraic framework, commutative and non-commutative, that is both powerful to use in applications and interesting to study in its own right. Naturally, this minicourse starts with the introduction of the shuffle product of words. We then recall some very simple properties of Riemann-Stieltjes integrals, and then directly move on to use them to show the shuffle relation, reparametrisation invariance and Chen's identity. We introduce the framework of the free Lie algebra and free Lie group in the context of the shuffle-deconcatenation Hopf algebra. This allows us to discuss varieties of signature tensors. The shuffle product splits into halfshuffles, which are really abstractions of Stieltjes integration. Halfshuffles satisfy the 'semiassociative' Zinbiel identity and allow us to understand how the signature behaves under polynomial transformations of the target space. Finally, we will see how we can describe the set of paths on a sphere (or on any affine variety) through the signature.

Registration is closed.