Tail asymptotics of the Brownian signature

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 C1, 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

Uniqueness of signature for simple curves

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1≤p<21≤p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLEκSLEκ null set, where 0<κ≤40<κ≤4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier transform of the n-point functions of SLE curves in terms of the expected signature of SLE curves. Although the techniques used in this article are deterministic, the results provide a platform for studying SLE curves through the signatures of their sample paths

The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied

A quasi-sure non-degeneracy property for the Brownian rough path

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-selfintersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization

Infant Cognitive Scores Prediction With Multi-stream Attention-based Temporal Path Signature Features

There is stunning rapid development of human brains in the first year of life. Some studies have revealed the tight connection between cognition skills and cortical morphology in this period. Nonetheless, it is still a great challenge to predict cognitive scores using brain morphological features, given issues like small sample size and missing data in longitudinal studies. In this work, for the first time, we introduce the path signature method to explore hidden analytical and geometric properties of longitudinal cortical morphology features. A novel BrainPSNet is proposed with a differentiable temporal path signature layer to produce informative representations of different time points and various temporal granules. Further, a two-stream neural network is included to combine groups of raw features and path signature features for predicting the cognitive score. More importantly, considering different influences of each brain region on the cognitive function, we design a learning-based attention mask generator to automatically weight regions correspondingly. Experiments are conducted on an in-house longitudinal dataset. By comparing with several recent algorithms, the proposed method achieves the state-of-the-art performance. The relationship between morphological features and cognitive abilities is also analyzed

Quasi-sure existence of gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons’ universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process

Uniform factorial decay estimates for controlled differential equations

We establish an uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations in the p-variation metric. As part of the proof, we also obtain a factorial decay estimate for controlled paths which is interesting in its own right