Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

中文翻译：

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金融市场的多重分形分析：综述

多重分形在复杂的自然和社会经济系统中无处不在。多重分形分析提供了强大的工具来理解不同领域时间序列的复杂非线性性质。受其与流体动力学湍流的惊人类比的启发，多重分形的概念起源于流体动力学湍流，金融市场的多重分形分析蓬勃发展，形成了经济物理学的主要方向之一。我们回顾了金融时间序列中采用或发明的多重分形分析方法和多重分形模型及其微妙的特性，它们适用于其他学科的时间序列。我们调查了在不同市场和不同时间段的金融时间序列中存在多重分形的累积证据，并讨论了多重分形的来源。介绍了多重分形分析在量化市场低效率、支持风险管理和开发其他应用程序方面的有用性。我们最后讨论了多重分形分析的开放问题和进一步的方向。