Meromorphic continuation of the mean signature of fractional brownian motion

TL;DR

This paper proves that the mean signature of multi-dimensional fractional Brownian motion can be meromorphically continued across the complex plane, revealing the structure and nature of its singularities related to hypergeometric integrals.

Contribution

It introduces a novel meromorphic continuation of the mean signature of fractional Brownian motion and characterizes the singularities through hypergeometric integrals and combinatorial structures.

Findings

Mean signature admits meromorphic continuation in the Hurst parameter.

Singularities are finite in order and located at rational numbers.

Each constituent integral is a sum of hypergeometric integrals.

Abstract

It is proved that the mean signature of multi-dimensional fractional brownian motion admits a meromorphic continuation in the hurst parameter to the entire complex plane. Each contstituent mean iterated integral is a sum of hypergeometric integrals indexed by the pair partitions which refine the partition arising from the sequential list of integrands which defines it. Furthermore, each such hypergeometric integral is holomorphic in the complement of a finite union of rational progressions determined by the combinatorial structure of the pair partition which defines it. It is not proved that these singularities actually exist, it is only proved that the singularities are of finite order and they can only occur in the specified discrete set of rational numbers.