[Paper Review] The Tail Asymptotics of the Brownian Signature

Horatio Boedihardjo, Xi Geng|arXiv (Cornell University)|Sep 26, 2016

Stochastic processes and financial applications10 references3 citations

TL;DR

This paper establishes a precise asymptotic relationship between the tail behavior of the Brownian signature and the quadratic variation of the path, extending Hambly and Lyons' result for $C^1$ paths to multidimensional Brownian motion. Using hyperbolic development and a novel subadditive estimate for signature asymptotics, it proves that the limsup of the normalized factorial norm of signature components converges to a constant $\kappa_d$, which characterizes the quadratic variation of Brownian motion.

ABSTRACT

The signature of a path γis 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 γwith its signature. If γis C^{1}, then an elegant formula of Hambly and Lyons relates the length of γto 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.

Motivation & Objective

• To extend the Hambly-Lyons formula for $C^1$ paths to multidimensional Brownian motion by identifying the role of quadratic variation in signature tail decay.
• To establish a precise asymptotic formula for the growth rate of the signature's higher-order components in the Brownian case.
• To develop a new subadditive estimate for signature asymptotics that may be of independent interest in rough path theory.
• To strengthen uniqueness results for Brownian rough paths by analyzing the tail behavior of their signatures.

Proposed method

• The authors use the hyperbolic development of Brownian motion to analyze its signature in a geometric setting of constant curvature $-1$.
• They derive a new subadditive estimate for the projective norm of signature components, which is central to the asymptotic analysis.
• The proof relies on the structure of iterated Itô integrals and the shuffle product identity for tensor algebras.
• The key technical step involves estimating the expected signature of Brownian motion using combinatorial arguments on word shuffles.
• The authors relate the growth rate of the signature to the quadratic variation of the path through the constant $\kappa_d$, derived from the limsup of normalized factorial norms.
• They use the fact that the lifting of Brownian motion to rough paths is almost surely well-defined, enabling almost sure results on the signature tail.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the tail of the Brownian signature in terms of its growth rate?
• RQ2 Can the quadratic variation of Brownian motion be recovered from the tail asymptotics of its signature?
• RQ3 Is there a subadditive estimate for signature norms that holds uniformly and can be used to derive such asymptotics?
• RQ4 Does the limsup of the normalized factorial norm of the signature components equal a universal constant for Brownian motion?
• RQ5 Can the uniqueness of Brownian rough paths be strengthened using tail signature behavior?

Key findings

• The limsup of $\left(\left(\frac{n}{2}\right)!\|\mathbb{B}_{0,1}^{n}\|_{\mathrm{proj}}\right)^{\frac{2}{n}}$ as $n \to \infty$ converges to $\kappa_d$ almost surely, where $\kappa_d$ is a constant depending on the dimension $d$.
• The constant $\kappa_d$ is shown to be related to the quadratic variation of the Brownian motion, playing a role analogous to path length in the $C^1$ case.
• A new subadditive estimate for the projective norm of signature components is derived, which is instrumental in proving the asymptotic result.
• The uniqueness of the Brownian rough path is strengthened: almost surely, no two distinct sample paths can have the same signature up to reparametrization, under a universal null set condition.
• The limsup in the asymptotic formula cannot be replaced by a supremum in the Brownian case, as the supremum may exceed the limsup with positive probability.
• The result holds for the entire Brownian rough path, and by projection to degree one, it also implies a uniqueness result at the level of sample paths.