From signature SDEs to affine and polynomial processes and back

Modern universal classes of dynamic processes, based on neural networks or signature methods, have recently entered the field of stochastic modeling, in particular in Mathematical Finance. This has opened the door to more data-driven and thus more robust model selection mechanisms, while first principles like no arbitrage still apply. We focus here on signature SDEs, i.e. (possibly Lévy driven) SDEs whose characteristics are linear functions of the process' signature and present methods how to learn these characteristics from data.
From a more theoretical point of view, we show how these new models can be embedded in the framework of affine and polynomial processes, which have been -- due to their tractability -- the dominating process class prior to the new era of highly overparametrized dynamic models. Indeed, we prove that generic classes of diffusion models can be viewed as infinite dimensional affine processes, which in this setup coincide with polynomial processes. A key ingredient to establish this result is again the signature process. This then allows to get power series expansions for expected values of analytic functions of the process' marginals, which also apply to signature SDEs.

Bio: Christa Cuchiero is professor at the University of Vienna. She received her Ph.D. in Mathematics from ETH Zurich in 2011. Her research interests include Mathematical Finance, Stochastic Analysis and Quantitative Risk Management, especially data driven risk inference, machine learning in Finance, multivariate stochastic and rough volatility modeling and stochastic portfolio theory. Christa was awarded several prizes and fellowships, in particular the START award of the Austrian Science Fund (FWF).