Abstract:
The theory of rough paths was introduced in Lyons (1998) as a way to understand and
overcome the lack of continuity of the Itô-map, which describes the dependence of the
solution of a stochastic differential equation (SDE) on the driving signal. Lyons observed
that by enhancing the driving process with a finite number of higher-order objects, one can
give a meaning in a pathwise sense to differential equations driven by
continuous rough signals, such as Brownian motion. These enhanced or lifted paths are called rough paths.
Furthermore, by equipping the space of rough paths with certain variation metrics, the Itô-map becomes continuous. These seminal ideas found many applications, in particular in stochastic analysis, providing a pathwise perspective and stability results to the theory of SDEs.
One important object in this theory is the signature of a path, whose original idea goes
back to Chen (1957, 1977) but later on, it has been taken up in the context of rough paths and introduced for more general rougher signals. The signature can be viewed as an enhancement of a (rough) path through iterated integrals. Its importance is particularly
due to its role as a well-suited feature map capturing essential path characteristics. Indeed, invarious respects, linear functionals of the signature act similarly to polynomials on path space and can be viewed as a canonical selection of basis functions. This explains why these signature based methods become more and more popular in machine learning, econometrics and mathematical finance.
While in its early development, the theory of rough paths primarily focused on continuous paths, advancements by Friz and Shekhar (2017) have extended this framework to include càdlàg paths and introduced a notion of signature that encompasses also paths with jumps.
Building on this definition, one main goal of this thesis is to rigorously establish the role
of the signature as a linear regression basis for path functionals in the càdlàg setting and to investigate the scope of its applicability from both theoretical and practical perspectives.
As a first step we prove a universal approximation theorem for continuous (with respect to
the J1 -Skorokhod topology) non-anticipative path functionals in terms of linear functionals
of the signature of càdlàg rough paths. Then, as an important application, we define a new class of universal signature models based on an augmented Lévy process, which we call
Lévy-type signature models. The approach that we follow consists in parameterizing the model itself or its semimartingale characteristics as linear functionals of the signature of an augmented Lévy process.
We show that these models extend the continuous signature models for asset prices as
proposed so far, while still preserving universality and tractability properties. Leveraging the universal approximation properties of the signature process, in the second part of the thesis we broaden the scope and introduce a more general class of jump-diffusions, which we call Sig-jump-diffusions, whose defining property is that the process semimartingale characteristics are parameterized by linear combinations of potentially infinitely many components of their own signature. For these processes, which includes the class of so-called holomorphic jump-diffusions introduced in the thesis, a significant extension of polynomial processes, we show that the expected signature, as well as the expected value of holomorphic functions of the process’ marginals, can be computed via duality methods for stochastic processes. In particular, these quantities admit ananalytic representation in terms of a solution of an (extended tensor algebra or sequence-valued) infinite-dimensional linear differential equation.
In the last part of the thesis, we derive a functional Itô-formula for non-anticipative maps
of rough paths building on the approximating properties of the signature and as a
byproduct of this result we show that sufficiently regular non-anticipative path functionals
admit a functional Taylor expansion.

Hybrid:
Hörsaal 2, Oskar-Morgenstern-Platz 1, 1090 Wien
Online:
https://univienna.zoom.us/j/66835936280?pwd=82ogNjbJPOsA8MzQyK8qGHtza4352A.1
Meeting ID: 668 3593 6280
Kenncode: 652439