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

  title={The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence},
  author={H. Boedihardjo and J. Diehl and M. Mezzarobba and H. Ni},
  journal={arXiv: Probability},
The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative. 
The expected signature of stopped Brownian motions on 2D domains has finite radius of convergence everywhere
A fundamental question in rough path theory asks if the expected signature of geometric rough paths completely determines the law of signature. One sufficient condition for the affirmative answer isExpand


Expected signature of Brownian Motion up to the first exit time from a bounded domain
The signature of a path provides a top down description of the path in terms of its effects as a control [Differential Equations Driven by Rough Paths (2007) Springer]. The signature transforms aExpand
Characteristic functions of measures on geometric rough paths
We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by itsExpand
The Signature of a Rough Path: Uniqueness
In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if andExpand
Stochastic area for Brownian motion on the Sierpinski gasket
We construct a Levy stochastic area for Brownian motion on the Sierpinski gasket. The standard approach via Ito integrals fails because this diffusion has sample paths which are far rougher thanExpand
On the signature and cubature of the fractional Brownian motion for H>12
Abstract We present several results concerning the fractional Brownian motion (fBm) for H > 1 ∕ 2 . First, we show that the rate of convergence of the expected signature of the linear piecewiseExpand
Convergence rates for the full Gaussian rough paths
Under the key assumption of finite {\rho}-variation, {\rho}\in[1,2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths areExpand
Cubature on Wiener space
  • Terry Lyons, Nicolas Victoir
  • Mathematics
  • Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2004
It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte CarloExpand
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications
Preface Introduction The story in a nutshell Part I. Basics: 1. Continuous paths of bounded variation 2. Riemann-Stieltjes integration 3. Ordinary differential equations (ODEs) 4. ODEs: smoothness 5.Expand
Hyperbolic development and inversion of signature
We develop a simple procedure that allows one to explicitly reconstruct any piecewise linear path from its signature. The construction is based on the development of the path onto the hyperbolicExpand
Elliptic Partial Differential Equa-tions of Second Order
Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's RepresentationExpand