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