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

## TL;DR

This paper investigates the properties of the expected signature of Brownian motion stopped at a boundary, providing a counterexample to a previously posed question about its moment conditions and convergence.

## Contribution

It presents the first example showing that the expected signature of Brownian motion may not satisfy the moment condition, challenging prior assumptions.

## Key findings

- Expected signature has finite radius of convergence
- Counterexample to the moment condition question
- Challenges previous beliefs about expected signatures

## 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.