Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content.

翻译：考虑到基于熵的市场信息与赫斯特指数都是判定有效市场假说是否适用于特定资产的有用工具，我们研究了这两种方法之间的关联。为此，当对数价格遵循分数布朗运动或其通过拉姆佩蒂变换得到的平稳扩展时，我们给出了市场信息的理论表达式。在后一种模型中，我们证明由于平稳性机制，赫斯特指数接近1/2可能导致时间序列具有极高的信息量。此外，我们引入了一种多尺度方法，以便根据信息集的大小对熵和市场信息进行更深入的解释。通过使用日度或日内数据对比特币、法国CAC 40指数、日经225指数以及欧元/美元汇率进行应用分析，验证了该方法论的内容。