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Discretely sampled signals and the rough Hoff process

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  • Flint, Guy
  • Hambly, Ben
  • Lyons, Terry

Abstract

We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead–lag increments. In particular, by sampling a d-dimensional continuous semimartingale X:[0,1]→Rd at a set of times D={ti}, we construct a piecewise linear, axis-directed process XD:[0,1]→R2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {Xt}ti∈D. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of XD. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong–Zakai Theorem (Wong and Zakai, 1965). Such random ODEs have a natural interpretation in the context of mathematical finance.

Suggested Citation

  • Flint, Guy & Hambly, Ben & Lyons, Terry, 2016. "Discretely sampled signals and the rough Hoff process," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2593-2614.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2593-2614
    DOI: 10.1016/j.spa.2016.02.011
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    References listed on IDEAS

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    1. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    2. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    3. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    4. Terry Lyons, 2014. "Rough paths, Signatures and the modelling of functions on streams," Papers 1405.4537, arXiv.org.
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    Citations

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    Cited by:

    1. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Numerical method for model-free pricing of exotic derivatives using rough path signatures," Papers 1905.01720, arXiv.org, revised Feb 2020.
    2. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Nonparametric pricing and hedging of exotic derivatives," Papers 1905.00711, arXiv.org.
    3. Imanol Perez Arribas & Cristopher Salvi & Lukasz Szpruch, 2020. "Sig-SDEs model for quantitative finance," Papers 2006.00218, arXiv.org, revised Jun 2020.
    4. Marc Sabate-Vidales & David v{S}iv{s}ka & Lukasz Szpruch, 2020. "Solving path dependent PDEs with LSTM networks and path signatures," Papers 2011.10630, arXiv.org.
    5. Fermanian, Adeline, 2021. "Embedding and learning with signatures," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    6. Hans Buhler & Blanka Horvath & Terry Lyons & Imanol Perez Arribas & Ben Wood, 2020. "A Data-driven Market Simulator for Small Data Environments," Papers 2006.14498, arXiv.org.

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