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 fairly simple. 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 recomendations are to read Friz and Hairer (2020), available free online, as an introduction, which covers the much-simpler Gaussian noise case.

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.

Signatures

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

References

Bonnier, Patric, Patrick Kidger, Imanol Perez Arribas, Cristopher Salvi, and Terry Lyons. 2019. “Deep Signature Transforms.” In Advances in Neural Information Processing Systems. Vol. 32. Curran Associates, Inc. http://arxiv.org/abs/1905.08494.
Chevyrev, Ilya, and Andrey Kormilitzin. 2016. “A Primer on the Signature Method in Machine Learning.” arXiv:1603.03788 [cs, Stat], March. http://arxiv.org/abs/1603.03788.
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. https://doi.org/10.1007/978-3-030-41556-3_1.
Hodgkinson, Liam, Fred Roosta, and Michael W Mahoney. 2021. “Stochastic Continuous Normalizing Flows: Training SDEs as ODEs.” Uncertainty in Artificial Intelligence 37 (July): 11.
Kalsi, Jasdeep, Terry Lyons, and Imanol Perez Arribas. 2020. “Optimal Execution with Rough Path Signatures.” SIAM Journal on Financial Mathematics 11 (2): 470–93. https://doi.org/10.1137/19M1259778.
Kelly, David. 2016. “Rough Path Recursions and Diffusion Approximations.” The Annals of Applied Probability 26 (1). https://doi.org/10.1214/15-AAP1096.
Kelly, David, and Ian Melbourne. 2014. “Smooth Approximation of Stochastic Differential Equations,” March. https://doi.org/10.1214/14-AOP979.
Lyons, Terry. 1994. “Differential Equations Driven by Rough Signals (I): An Extension of an Inequality of L. C. Young.” Mathematical Research Letters 1 (4): 451–64. https://doi.org/10.4310/MRL.1994.v1.n4.a5.
———. 2014. “Rough Paths, Signatures and the Modelling of Functions on Streams.” arXiv:1405.4537 [math, q-Fin, Stat], May. http://arxiv.org/abs/1405.4537.
Lyons, Terry J., and Nadia Sidorova. 2005. “Sound 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. https://www.ucl.ac.uk/~ucahnsi/Papers/lyons_sidorova_capetown.pdf.
Twardowska, Krystyna. 1996. “Wong-Zakai Approximations for Stochastic Differential Equations.” Acta Applicandae Mathematica 43 (3): 317–59.

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