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A Representation for the Expected Signature of Brownian Motion up to the First Exit Time of the Planar Unit Disc

In: Stochastic Analysis and Applications 2025

Author

Listed:
  • Horatio Boedihardjo

    (University of Warwick, Department of Statistics)

  • Lin Hao He

    (University of Oxford, Mathematical Institute)

  • Lisa Wang

    (TU Wien, Institute of Discrete Mathematics and Geometry)

Abstract

The signature of a sample path is a formal series of iterated integrals along the path. The expected signature of a stochastic process gives a summary of the process that is especially useful for studying stochastic differential equations driven by the process. Lyons-Ni derived a partial differential equation for the expected signature of Brownian motion, starting at a point z in a bounded domain, until it hits the boundary of the domain. We focus on the domain of planar unit disc centred at 0. Motivated by recently found explicit formulae for the expected hyperbolic development of this process in terms of Bessel functions, we derive a tensor series representation for this expected signature, coming from studying Lyons-Ni’s PDE. Although the representation is rather involved, it simplifies significantly to give a formula for the polynomial leading degree term in each tensor component of the expected signature.

Suggested Citation

  • Horatio Boedihardjo & Lin Hao He & Lisa Wang, 2026. "A Representation for the Expected Signature of Brownian Motion up to the First Exit Time of the Planar Unit Disc," Springer Books, in: Dan Crisan & Ilya Chevyrev & Thomas Cass & James Foster & Christian Litterer & Cristopher Salvi (ed.), Stochastic Analysis and Applications 2025, pages 189-213, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-03914-9_7
    DOI: 10.1007/978-3-032-03914-9_7
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