Statistical Learning with Path Signatures: From Universal Approximation to Derivative Hedging

张瑞勋，北京大学数学科学学院金融数学系主任，研究员/长聘副教授，统计科学中心、大数据分析与应用技术国家工程实验室、数量经济与数理金融教育部重点实验室成员。2011年获北京大学数学与应用数学、经济学（双学位）学士；2015年获麻省理工学院（MIT）应用数学博士。近年研究兴趣集中在AI+Finance，入选国家海外高层次人才计划青年项目，主持教育部U40项目、国家重点研发计划青年科学家项目、国家自然科学基金面上项目等，出版专著3部，在PNAS、MS、OR、JASA、MF等国际顶尖期刊上发表论文30余篇，研究工作近5年获INFORMS和SIAM等国际最佳论文奖7项。任Mathematical Finance等期刊副主编，获2025年Operations Research审稿人杰出服务奖。 

Path signatures—the hierarchy of iterated integrals of a path — provide a canonical way to turn nonlinear functionals of time series into linear regressions, justified by a universal approximation theorem. This theorem, however, is only an existence result. This talk presents a line of work that closes three gaps separating it from a usable methodology in finance. First, an approximation-rate theory: for smooth functionals of Itô diffusions, the truncation error of level-K signature approximations decays at a minimax-sharp polynomial rate, so signatures act as polynomials on path space. Second, an estimation theory: we characterize when Lasso consistently selects signature features, showing that the choice between Itô and Stratonovich signatures is a statistical decision that depends on the underlying dynamics. Third, a tradability theory: every discretized Itô signature is the PnL of a self-financing trading strategy, so a single fitted linear representation of a payoff simultaneously prices (by expectation) and hedges (by replication)—yielding an interpretable, model-free hedging framework with end-to-end error guarantees and robust performance on fifteen years of S&P 500 options. Based on joint works with Xin Guo, Blanka Horvath, Wen Su, Wu Su, Binnan Wang, and Chaoyi Zhao.