Electronic Communications in Probability

Signature inversion for monotone paths

Jiawei Chang, Nick Duffield, Hao Ni, and Weijun Xu

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The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every $N$, we sample a lattice path of $N$ steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a “wrong” path is exponentially small in $N$. The argument relies on a probabilistic interpretation of the signature for monotone paths.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 42, 11 pp.

Received: 8 February 2017
Accepted: 23 June 2017
First available in Project Euclid: 15 August 2017

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Primary: 60

signature inversion monotone paths

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Chang, Jiawei; Duffield, Nick; Ni, Hao; Xu, Weijun. Signature inversion for monotone paths. Electron. Commun. Probab. 22 (2017), paper no. 42, 11 pp. doi:10.1214/17-ECP70. https://projecteuclid.org/euclid.ecp/1502762748

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  • [1] Dembo, A. and Zajic, T.: Large deviations: From empirical mean and measure to partial sums process. Stoch. Proc. Appl., 57, (1995), 191–224.
  • [2] Dembo, A. and Zeitouni, O.: Large Deviations Techniques and Applications. Springer, (1998).
  • [3] Duffy, K. and Rodgers-Lee: Some useful functions for functional large deviations. Stochastics and Stochastic Reports 76, no.3, (2004), 267–279.
  • [4] Hambly, B. and Lyons, T.: Some notes on trees and paths. arXiv:math.0809.1365.
  • [5] Hambly, B. and Lyons, T.: Uniqueness for the signature of a path of bounded variation and the reduced path group. Ann. Math., 171, no.1, (2010), 109–167.
  • [6] Lyons, T. and Xu, W.: Inverting the signature of a path. J. Eur. Math. Soc., to appear.