Abstract: Signatures are iterated path integrals of continuous and discrete-time processes, and their universal nonlinearity linearizes the problem of feature selection in time series data analysis. This paper studies the consistency of signature using Lasso regression, both theoretically and numerically. We establish conditions under which the Lasso regression is consistent both asymptotically and in finite sample. Furthermore, we show that the Lasso regression is more consistent with the Itô signature for time series and processes that are closer to the Brownian motion and with weaker inter-dimensional correlations, while it is more consistent with the Stratonovich signature for mean-reverting time series and processes. We demonstrate that signature can be applied to learn nonlinear functions and option prices with high accuracy, and the performance depends on properties of the underlying process and the choice of the signature. This is joint work with Xin Guo, Binnan Wang, and Chaoyi Zhao.

Bio: Ruixun Zhang is an assistant professor and Boya Young Fellow in the Department of Financial Mathematics, School of Mathematical Sciences at Peking University (PKU). He is currently a visiting scholar at Columbia University in the IEOR department in Fall 2024. Ruixun received a PhD in applied mathematics from MIT in 2015, and bachelor’s degrees in Mathematics and Applied Mathematics, and Economics (double degree) from Peking University in 2011. Ruixun’s research interests include machine learning, green finance, and evolutionary models of financial behavior. His research has appeared in journals such as PNAS, Management Science, and Operations Research. His work has been recognized by the S&P Global ESG Academic Research Award, ICPM Academic Research Award, CEMA Best Paper Award, and CFRI&CIRF Best Paper Award.