The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

Horatio Boedihardjo,Hao Ni,Joscha Diehl,Marc Mezzarobba

doi:10.1112/blms.12420

Abstract

The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.

Highlights

The expected signature is an analogue of the Laplace transform for probability measures on rough paths
The iterated-integrals signature S(X)0,T defined by the expression (1) — using the given integration theory — is a random variable
In this work we show that the radius of convergence is finite

Summary

Differentiability of the development of expected signature

We first need two technical lemmas which assert that the development of the expected signature is twice differentiable, and satisfies the PDE we expect it to. The function z → projn(Φ(z)) is twice continuously differentiable. By [8, Theorem 2.2], which bounds the values of a function u in terms of the Sobolev norm of u, there is some constant C(2) such that for all z ∈ D and |α| 2,. Since M projn(Φ(z)) is a linear image of projn(Φ(z)), the function M projn(Φ(z)) is twice continuously differentiable in z, and there exists c > 0 such that for all z ∈ D,. This bound allows us to deduce that for |α| = 2, the series λnDαM projn (Φ(z)). By Lemma 1, each infinite sum converges and we may take the derivatives outside the infinite sum

A polar decomposition for the development

Conclusion