arXiv subjects

Andrew Ursitti

Publications and source records attributed to Andrew Ursitti.

2 recordsLinked to original sources

arXiv2018

Meromorphic continuation of the mean signature of fractional brownian motion

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.

math.PR↗

arXiv2016

Computation of some transcendental integrals from path signatures

It is shown that if $γ$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $γ$ then $s\mapsto F(s)=\int_γf^sg dh$ defines an entire holomorphic function provided the convex hull of the image of $f$ does not contain zero. If in addition $|\log z|\leq \log 2$ on the convex hull of the image of $f$ then for any $s\in \mathbf{C}$, $F(s)$ can be computed from the nonnegative integer values $\{F(k)\}_{k\in \mathbf{N}}$. If in addition to these hypotheses each of $f,g,h$ is a polynomial, then the values $F(k)$ are computable directly from the signature of $γ$ thus all values of $F(s)$ are computable from the signature. As a special case the winding number of a closed path $γ$ around an affine submanifold of codimension two is computed from finitely many terms of the signature provided certain estimates are satisfied.

math.CA↗