Rough path theory, differential equations and stochastic analysis View Homepage


Ontology type: schema:MonetaryGrant     


Grant Info

YEARS

2010-2016

FUNDING AMOUNT

850820.0 EUR

ABSTRACT

We propose to study stochastic (classical and partial) differential equations and various topics of stochastic analysis, with particular focus on the interplay with T. Lyons' rough path theory: 1) There is deep link, due to P. Malliavin, between the theory of hypoelliptic second order partial differential operators and certain smoothness properties of diffusion processes, constructed via stochastic differential equations. There is increasing evidence (F. Baudoin, M. Hairer &) that a Markovian (=PDE) structure is dispensable and that Hoermander type results are a robust feature of stochastic differential equations driven by non-degenerate Gaussian processes; many pressing questions have thus appeared. 2) We return to the works of P.L. Lions and P. Souganidis (1998-2003) on a path-wise theory of fully non-linear stochastic partial differential equations in viscosity sense. More specifically, we propose a rough path-wise theory for such equations. This would in fact combine the best of two worlds (the stability properties of viscosity solutions vs. the smoothness of the Ito-map in rough path metrics) to the common goal of the analysis of stochastic partial differential equations. On a related topic, we have well-founded hope that rough paths are the key to make the duality formulation for control problems a la L.C.G. Rogers (2008) work in a continuous setting. 3) Rough path methods should be studied in the context of (not necessarily continuous) semi-martingales, bridging the current gap between classical stochastic integration and its rough path counterpart. Related applications are far-reaching, and include, as conjectured by J. Teichmann, Donsker type results for the cubature tree (Lyons-Victoir s powerful alternative to Monte Carlo). More... »

URL

http://cordis.europa.eu/project/rcn/95481_en.html

Related SciGraph Publications

  • 2016-12-03. Eikonal equations and pathwise solutions to fully non-linear SPDEs in STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS: ANALYSIS AND COMPUTATIONS
  • 2016-08-12. On the existence of SLE trace: finite energy drivers and non-constant κ in PROBABILITY THEORY AND RELATED FIELDS
  • 2016-02-11. Stochastic control with rough paths in APPLIED MATHEMATICS & OPTIMIZATION
  • 2015-05-08. Random Matrices in Non-confining Potentials in JOURNAL OF STATISTICAL PHYSICS
  • 2014-08-20. On a Chen–Fliess approximation for diffusion functionals in MONATSHEFTE FÜR MATHEMATIK
  • 2014-07-12. Tracy–Widom at High Temperature in JOURNAL OF STATISTICAL PHYSICS
  • 2013-02-08. Rough path stability of (semi-)linear SPDEs in PROBABILITY THEORY AND RELATED FIELDS
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