The Rough Path Associated to the Multidimensional Analytic Fbm with Any Hurst Parameter

Abstract. In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by Γ. This process has been introduced in [16], and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on Γ can be constructed for any value of the Hurst parameter in (0, 1/2). This allows in particular to solve differential equations driven by Γ in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli [6].