  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 It\^o 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.
