The Tail Asymptotics of the Brownian Signature
Abstract
The signature of a path γ is a sequence whose n-th term is the order-n iterated integrals of γ. It arises from solving multidimensional linear differential equations driven by γ. We are interested in relating the path properties of γ with its signature. If γ is C1, then an elegant formula of Hambly and Lyons relates the length of γ to the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion, with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion.
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