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