Periodic Reporting for period 4 - GPSART (Geometric aspects in pathwise stochastic analysis and related topics)
Periodo di rendicontazione: 2021-03-01 al 2022-08-31
(i) significantly furthered the transfer of concepts from rough path theory to - and from - the world of Hairer's regularity structures;
(ii) found some decisive applications of geometric and pathwise ideas in quantitative finance;
(iii) initiated the use of rough path-inspired analytic ideas in the world of conformally invariant processes (Schramm-Loewner evolution) and
(iv) exhibited new solution strategies for the pathwise analysis of non-linear evolution equations.
(ii) Applications of geometric and pathwise ideas in quantitative finance were investigated, more specifically related to a new paradigm n financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. Our main insight is that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models. We expect to use this machinery in various ways, notably towards derivations of asymptotic pricing formulae. Interestingly, some of the mathematical ideas here, like large deviation asymptotics, have precise parallels in the world of singular SPDEs studied amongst others in (i).
(iii) Concerning a pathwise understanding of the geometry of Loewner evolution, we revisited regularity of SLE trace, for all kappa, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem rooted in the Garsia--Rodemich--Rumsey lemma, we obtained in particular finite moments (and hence almost surely) optimal variation regularity with index min(1+kappa/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler. We could push this method to extend previous continuity results of SLE by Rohde and coworkers.
At last, concerning (iv), we have studied the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic partial differential equations with quadratic Hamiltonians associated to a Riemannian geometry. The results extend the class of equations studied previously by Lions and Souganidis.
A number of results were obtain outside, if loosely inspired, the original project structure. This includes some works on rough paths and mean-field interacting particle systems, rough paths with jumps and their successful applications to homogenization and fast-slow systems. Foundational work on stochastic rough equations, also related to the new notion of rough semimartingale, was seen to have a surprising impact on some pathwise SPDEs seen in (iv).
Concerning dissemination, the PI and his research group organized two major international conferences and 13 workshops devoted to the topics of this project, and as prime opportunity to discuss related progress.