Periodic Reporting for period 4 - GPSART (Geometric aspects in pathwise stochastic analysis and related topics)
Reporting period: 2021-03-01 to 2022-08-31
The starting point of this project was 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). In a spectacular development, the theory of such ``rough paths" has been extended to ``rough fields", notably in the astounding works of M. Hairer (from '13 on). Within this project we have
(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.
(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.
(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 and precise Laplace asymptotics (refinements of previous large deviations results of Haier and Weber), 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 the geometric, quasi-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. The algebraic aspects of our investigation triggered an interesting spin-off developments in algebraic geometry. An unexpected relation between KPZ type expansions, rough volatility and cumulants was discovered, also related to expected signatures.
(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.
(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.