Tail asymptotics of the Brownian signature

Boedihardjo, H.; Geng, X.

doi:10.1090/tran/7683

Tail asymptotics of the Brownian signature
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by H. Boedihardjo and X. Geng PDF

Trans. Amer. Math. Soc. 372 (2019), 585-614 Request permission

Abstract:

The signature of a path $\gamma$ is a sequence whose $n$-th term is the order-$n$ iterated integrals of $\gamma$. It arises from solving multidimensional linear differential equations driven by $\gamma$. We are interested in relating the path properties of $\gamma$ with its signature. If $\gamma$ is $C^{1}$, then an elegant formula of Hambly and Lyons relates the length of $\gamma$ 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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