Li, Siran; Ni, Hao; (2022) Expected signature of stopped Brownian motion on d-dimensional C2,α-domains has finite radius of convergence everywhere: 2≤d≤8. Journal of Functional Analysis , 282 (15) , Article 109447. 10.1016/j.jfa.2022.109447.

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Abstract

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain Ω, it is only known that the radius of convergence for the expected signature on sufficiently regular Ω is strictly positive everywhere, and that the radius of convergence is finite at some point when Ω is the 2-dimensional unit disc ([1]). In this paper, we prove that on any bounded C2,α-domain Ω⊂ℝd with 2≤d≤8, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.

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