Is it possible to compare Rough path theory and White noise Theory?

Question

Please accept apology if this question is vague. (Would you please comment rather then downvote, I may be stopped to ask more questions. I will delete my question if required.)

It is related to the link which describes white noise theory to deal with stochastic differential equation. https://www.duo.uio.no/bitstream/handle/10852/10633/pm02-03.pdf?sequence=1&isAllowed=y On the other hand another theory relates to the same area as described in Wikipedia. https://en.wikipedia.org/wiki/Rough_path

Both of them shows that iterated integrals (different measures) has uniqueness. Rough path theory is very popular as it has has been used in machine learning however, white noise theory is not that popular and it seems it has not been used as extensively as the other. Is it possible to have comparison between the two? Any comments would be highly appreciated. Any reference would also be very helpful. I am looking for some insight to compare this two approach.

Answer 1

I looked briefly at the paper you linked - it looked to be about Malliavin calculus.

Yes, both Malliavin calculus and rough path theory have these iterated integrals. However what those integrals mean are slightly different. In Malliavin calculus those iterated integrals are usually defined probabilistically - i.e. like a Ito integral. In rough paths the integrals are defined pathwise and are postulated.

However, in the case of fractional Brownian motion the way we define the integrals in rough path theory is through Malliavin calculus.

It goes like this:

-Use Malliavin calculus to define iterated integral as a stochastic integral

-Using your definition of stochastic integral above, you define the values of pathwise integrals in rough path theory

Answer 2

I think the best reference for you is the course of Hairer and Friz http://www.hairer.org/notes/RoughPaths.pdf where they deal with both ito integration and Stratonovich integration.