标题： 分数布朗运动的路径签名正则化学习 显示英文标题

标题： Regularized Learning for Fractional Brownian Motion via Path Signatures

摘要： 分数布朗运动（fBm）通过引入由Hurst参数 $H\in (0,1)$ 控制的增量之间的依赖关系扩展了经典布朗运动。与传统布朗运动不同，fBm的增量不是独立的。由分数布朗运动生成的路径可能表现出显著的不规则性，尤其是在Hurst参数较小时。因此，经典的回归方法可能无法有效发挥作用。签名，定义为连续和离散时间过程的迭代路径积分，具有普遍的非线性特性，通过有效地将其线性化，简化了时间序列数据分析中的特征选择挑战。因此，在处理不规则数据时，我们采用套索回归技术进行正则化。为了评估签名套索回归在分数布朗运动（fBM）上的性能，我们研究了当Hurst参数 $ H \ne \frac{1}{2} $ 时的一致性，这涉及推导签名的一阶和二阶矩的界限。对于 $ H > \frac{1}{2} $ 的情况，我们使用Young意义下的签名，而对于 $ H < \frac{1}{2} $ 的情况，则使用Stratonovich解释。仿真结果表明，签名套索回归在合成数据以及真实世界的数据集上可以优于传统的回归方法。 显示英文摘要

摘要： Fractional Brownian motion (fBm) extends classical Brownian motion by introducing dependence between increments, governed by the Hurst parameter $H\in (0,1)$. Unlike traditional Brownian motion, the increments of an fBm are not independent. Paths generated by fractional Brownian motions can exhibit significant irregularity, particularly when the Hurst parameter is small. As a result, classical regression methods may not perform effectively. Signatures, defined as iterated path integrals of continuous and discrete-time processes, offer a universal nonlinearity property that simplifies the challenge of feature selection in time series data analysis by effectively linearizing it. Consequently, we employ Lasso regression techniques for regularization when handling irregular data. To evaluate the performance of signature Lasso on fractional Brownian motion (fBM), we study its consistency when the Hurst parameter $ H \ne \frac{1}{2} $. This involves deriving bounds on the first and second moments of the signature. For the case $ H > \frac{1}{2} $, we use the signature defined in the Young sense, while for $ H < \frac{1}{2} $, we use the Stratonovich interpretation. Simulation results indicate that signature Lasso can outperform traditional regression methods for synthetic data as well as for real-world datasets.