Fractional signature: a generalisation of the signature inspired by fractional calculus

The signature of a path is a sequence of integrals, applied iteratively to the components of the path, which allows it to describe the path precisely and to summarise its characteristics, especially the geometrical ones. This concept emerged in the 1950s and was originally studied by K. T. Chen, who developed the theory and gave the first significant results that justified the interest in the signature. After that, the signature formed part of the Terry Lyons' theory of rough paths, which is key in the field of stochastic calculus and in the study of differential equations controlled by rough paths. Thanks to the development of this theory, the signature was generalised to apply to certain paths of finite variation and regained some relevance. More recently, applications of the signature have also been found in the field of machine learning, where its properties for describing paths are useful for summarising the data sequences used in this discipline and for revealing their properties, which facilitates the training of models that have to make decisions based on the data. Before we begin, we start by defining some concepts.