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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - GPSART (Geometric aspects in pathwise stochastic analysis and related topics)

Teaser

Summary

\"Recent years have seen an explosion of applications of geometric and pathwise ideas in probability theory, with motivations from fields as diverse as quantitative finance, statistics, filtering, control theory and statistical physics. Much can be traced back to Bismut, Malliavin (1970s) on the one-hand and then Doss, Sussman (1970s), Foellmer (1980s) on the other hand, with substantial new input from Lyons (from \'94 on), followed by a number of workers, including Gubinelli (from \'04 on) and the PI of this project (also from \'04 on). Most recently, the theory of such ``rough paths\"\" has been extended to ``rough fields\"\", notably in the astounding works of M. Hairer (from \'13 on). The purpose of this project is to study a number ofimportant problems in this field, going beyond the rough path setting, and with emphasis on geometric ideas.

(i) The transfer of concepts from rough path theory to the new world of Hairer\'s regularity structures.
(ii) Applications of geometric and pathwise ideas in quantitative finance.
(iii) Obtain a pathwise understanding of the geometry of Loewner evolution and more generally explore the use of rough
path-inspired ideas in the world of Schramm-Loewner evolution.
(iv) Investigate the role of geometry in the pathwise analysis of non-linear evolution equations.\"

Work performed

\"Work performed from the beginning of the project to the end of the period covered by the report with main results achieved so far and progress beyond the state of the art / expected results until the end of the project

(i) We studied the transfer of concepts from rough path theory to the new world of Hairer\'s regularity structures. This includes Malliavin calculus and support theorems for singular stochastic partial differential equations, in the spirit of previous works for rough differential equations. A Rough Path Perspective on Renormalization was introduced: we revisit (higher-order) translation operators on rough paths, in both the geometric and branched setting. As in Hairer\'s work on the renormalization of singular SPDEs we propose a purely algebraic view on the matter. Recent advances in the theory of regularity structures, especially the Hopf algebraic interplay of positive and negative renormalization of Bruned--Hairer--Zambotti (2016), are seen to have precise counterparts in the rough path context, even with a similar formalism (short of polynomial decorations and colourings). Renormalization is then seen to correspond precisely to (higher-order) rough path translation. Many other \"\"transfer question\"\", as outlined in the proposal, remain.

(ii) Applications of geometric and pathwise ideas in quantitative finance were investigated. A new paradigm recently emerged in 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.

(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 of Garsia--Rodemich--Rumsey type, we obtain 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) HÃ¶lder regularity Ã la Johansson Viklund and Lawler. We expect to derive further stability results for SLE until the end of the project

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.\"