A plethora of stochastic models used in particular in mathematical finance, but also population genetics and physics, stems from the class of affine and polynomial processes. The history of these processes is on the one hand closely connected with the important concept of tractability, that is a substantial reduction of computational efforts due to special structural features, and on the other hand with a unifying framework for a large number of probabilistic models. One early instance in the literature where this unifying affine and polynomial point of view can be applied is Lévy's stochastic area formula. Starting from this example, we present a guided tour through the main properties and recent results, which lead to signature stochastic differential equations (SDEs). They constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are entire or real-analytic functions of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in subsets of group-like elements of the extended tensor algebra. Signature SDEs are thus a class of stochastic processes, which is universal within Itô processes with path-dependent characteristics and which allows - due to the affine theory - for a relatively explicit characterization of the Fourier-Laplace transform and hence the full law on path space.
Christa Cuchiero is a professor at the University of Vienna. She earned her doctorate in Mathematics from ETH Zurich in 2011. Her research centres around mathematical finance, stochastic analysis, quantitative risk management and machine learning. She is particularly interested in classes of universal stochastic processes with applications in volatility modelling and portfolio theory, approximation theory in dynamic situations, data-driven risk inference and machine learning in finance. Christa Cuchiero has received several prizes and fellowships, including the prestigious START award of the Austrian Science Fund (FWF). She has given a number of keynote speeches and serves on the editorial board of several academic journals. She has also co-organized international conferences and a world online seminar series on machine learning in finance.