On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index

In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the $(\alpha, \beta)$-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.