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.

翻译：签名是连续与离散时间过程的迭代路径积分，其通用非线性性质将时间序列数据分析中的特征选择问题线性化。本文从理论与数值两方面研究了基于Lasso回归的签名一致性。我们建立了Lasso回归在渐近和有限样本条件下均具有一致性的理论条件。进一步研究表明：对于更接近布朗运动且维度间相关性更弱的时间序列与过程，Lasso回归与Itô签名具有更高的一致性；而对于均值回复型时间序列与过程，则与Stratonovich签名的一致性更优。我们证明签名可高精度地应用于非线性函数学习与期权定价，其性能取决于底层过程的特性与签名的选择。