Rough path theory

Also, signatures.

I am not sure yet. Some kind of alternative extension of integrals which happens to make pathwise calculations ove stochastic integrals simple, in some sense. I am pretty sure they mean rough in the sense of approximate rather than the sense of not smooth. Or maybe both?

Seems to originate in a fairly impenetrable body of work by Lyons, e.g. T. Lyons (1994) but modern recommendations are to read Friz and Hairer (2020), available free online, as an introduction, which covers the simplest (?) case of Gaussian noise.

Discrete approximations

Wong-Zakai approximations Twardowska (1996). (Martin Hairer recommendation.)

Possibly compact refs: (Kelly 2016; Kelly and Melbourne 2014).

In learning

Hodgkinson, Roosta, and Mahoney (2021) makes use of rough path integrals to justify learning by the adjoint method in stochastic differential equations.


Chevyrev and Kormilitzin (2016) discusses path signatures in particular, which is something arising in the theory about which I know little.


Bonnier, Patric, Patrick Kidger, Imanol Perez Arribas, Cristopher Salvi, and Terry Lyons. 2019. β€œDeep Signature Transforms.” In Advances in Neural Information Processing Systems. Vol. 32. Curran Associates, Inc.
Chevyrev, Ilya, and Andrey Kormilitzin. 2016. β€œA Primer on the Signature Method in Machine Learning.” arXiv:1603.03788 [Cs, Stat], March.
Friz, Peter K., and Martin Hairer. 2020. A Course on Rough Paths. Edited by Peter K. Friz and Martin Hairer. Universitext. Cham: Springer International Publishing.
Hodgkinson, Liam, Fred Roosta, and Michael W Mahoney. 2021. β€œStochastic Continuous Normalizing Flows: Training SDEs as ODEs.” Uncertainty in Artificial Intelligence 37 (July): 11.
Kalsi, Jasdeep, Terry Lyons, and Imanol Perez Arribas. 2020. β€œOptimal Execution with Rough Path Signatures.” SIAM Journal on Financial Mathematics 11 (2): 470–93.
Kelly, David. 2016. β€œRough Path Recursions and Diffusion Approximations.” The Annals of Applied Probability 26 (1).
Kelly, David, and Ian Melbourne. 2014. β€œSmooth Approximation of Stochastic Differential Equations,” March.
Kidger, Patrick. 2022. β€œOn Neural Differential Equations.” Oxford.
Lyons, Terry. 1994. β€œDifferential Equations Driven by Rough Signals (I): An Extension of an Inequality of L. C. Young.” Mathematical Research Letters 1 (4): 451–64.
β€”β€”β€”. 2014. β€œRough Paths, Signatures and the Modelling of Functions on Streams.” arXiv:1405.4537 [Math, q-Fin, Stat], May.
Lyons, Terry J., and Nadia Sidorova. 2005. β€œSound Compression: A Rough Path Approach.” In Proceedings of the 4th International Symposium on Information and Communication Technologies, 223–28. WISICT ’05. Cape Town, South Africa: Trinity College Dublin.
Twardowska, Krystyna. 1996. β€œWong-Zakai Approximations for Stochastic Differential Equations.” Acta Applicandae Mathematica 43 (3): 317–59.

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