Stochastic processes with non-zero higher order variations

Question

I'm under the impression that how non-zero quadratic variation of the Brownian motion results in Itō's lemma or in general, the creation of the Itō's calculus. I'm also aware that stochastic integral can also be generalized to martingales. I'd like to know that

Has the generalized stochastic calculus to stochastic processes with non-zero higher order variations been studied? if so, are they of any use?

Answer 1

here are my two cents

Specifically you can look at fractional Brownian motion with Hurst index different from 1/2. There is a stochastic integral and a stocahstic calculus for those processes but I believe this is less general than what you are looking for.

I know otherwise that Young integrals allows integrators with q-finite variations in the deterministic case and that maybe rough path analysis can be be used to define stochastic integrals that might if not mistaken match what you are looking for.

Anyway intrinsically as Bichteler-Dellacherie's theorem demonstrate, semi-martingales (in a nutshell Quadratic variation processes + FV processes) are the best processes to define a stochastic integral in the sense of the existence of a "dominated convergence theorem" inside, if I may say.

Best regards.