Papers › Expected signature of stopped Brownian motion on d-dimensional C^(2, α)-domains has...
Expected signature of stopped Brownian motion on d-dimensional C^(2, α)-domains has finite radius of convergence everywhere: 2≤d ≤8
Siran Li, Hao Ni
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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 C^(2,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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