In complex systems, crucial parameters are often subject to unpredictable changes in time. Climate, biological evolution and networks provide numerous examples for such non-stationarities. In many cases, improved statistical models are urgently called for. In a general setting, we study systems of correlated quantities to which we refer as amplitudes. We are interested in the case of non-stationarity, i.e., seemingly random covariances. We present a general method to derive the distribution of the covariances from the distribution of the amplitudes. To ensure analytical tractability, we construct a properly deformed Wishart ensemble of random matrices. We apply our method to financial returns where the wealth of data allows us to carry out statistically significant tests. The ensemble that we find is characterized by an algebraic distribution which improves the understanding of large events.
Correction to "Leverage and volatility feedback effects in high-frequency data" [J. Financial Econometrics 4 (2006) 353--384]
Bollerslev et al.（2006）研究了Heston（1993）随机波动率模型下平方收益率的互协方差。为了获得这些互协方差，作者使用了一个不正确的平方收益分布表达式。在这里，我们将获得平方收益率的正确分布，并检查在这个新分布下，Bollerslev et al.（2006）附录A.2中的结果仍然成立。
Physics-Informed Singular-Value Learning for Cross-Covariances Forecasting in Financial Markets
Université Paris-Saclay, CentraleSupélec, Mathématiques et Informatique pour la Complexité et les Systèmes（巴黎-萨克雷大学，CentraleSupélec，复杂性与系统数学与计算机科学）；Università degli Studi di Palermo, Dipartimento di Fisica e Chimica "Emilio Segrè"（巴勒莫大学，Emilio Segrè物理与化学系）
SBBTS: A Unified Schrödinger-Bass Framework for Synthetic Financial Time Series
BNPP and Ecole Polytechnique（法国巴黎银行和巴黎综合理工学院）；BNPP and Monash University（法国巴黎银行和莫纳什大学）；BNPP and ENSAE Paris（法国巴黎银行和巴黎高等经济商业学院）；LPSM, Université Paris Cité and Sorbonne University（巴黎西岱大学LPSM实验室和索邦大学）；Ecole Polytechnique, CMAP（巴黎综合理工学院CMAP实验室）
