Minerva Foundation Lectures
Tuesdays 4:10–5:25 PM
Room 507 Mathematics
Professor Rama Cont
Oxford University
Rough Calculus
The Ito calculus may be viewed as an extension of the Newton-Leibniz calculus to
smooth functions of paths with non-zero quadratic variation. This analytical
viewpoint is exploited to develop a calculus for smooth function(al)s of irregular
paths with non-zero p-th variation for arbitrary p>1. Although this “rough
calculus” is strictly pathwise in nature and does not involve any probabilistic
ingredient, it is applicable to stochastic processes with irregular paths.
We illustrate the concepts and results of this theory in the setting of the Ito-
Föllmer calculus for smooth function(al)s of paths with finite quadratic variation.
We will then show how these results may be extended to the more general setting
of smooth functionals of paths with non-zero p-th variation for arbitrary p>1,
leading to a higher order Ito-type calculus. Finally, we will sketch some examples of
applications to transport equations, optimal control and rough dynamics on
manifolds.
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