Probability and Mathematical Physics Seminar

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Rough calculus is a pathwise extension of the Ito calculus to smooth function(al)s of paths and processes with arbitrary Hölder regularity.
We construct integrals of k-jets along irregular paths with finite p-th variation for any p >1, and derive change of variable formulas for smooth functions of such paths. The case p=2  corresponds to Hans Foellmer's pathwise Ito formula while the case p>2 leads to a "higher order Ito calculus".
The framework is entirely pathwise but applicable to stochastic processes with irregular trajectories, such as (fractional) Brownian motions with arbitrary Hurst exponent. As an application, we show how these results lead to new insights on the transport of measures along rough trajectories.