Wednesday, May 8, 2019 - 14:00 in V3-201
Rough calculus: pathwise integration and non-anticipative calculus for functionals of irregular paths
A talk in the Bielefeld Stochastic Afternoon series by
Rama Cont from University of Oxford
| Abstract: |
We construct a pathwise calculus which extends the Ito calculus to smooth functionals of continuous paths with regularity defined in terms of the $p$-th variation along a sequence of time partitions for arbitrary large $p>0$. We show pointwise convergence of appropriately defined compensated Riemann sums. The corresponding pathwise integral satisfies a change of variable formula and an isometry formula. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. Finally, we obtain a "signal plus noise" decomposition for regular functionals of paths with strictly increasing $p$-th variation. Our results apply to sample paths of semimartingales as well as fractional Brownian motions with arbitrary Hurst parameter $H>0$.
Based on joint work with: Anna Ananova (Oxford) and Nicholas Perkowski (Berlin).
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