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End date: 2019 × Start date: 2010 × DDC: 510 × Subject: Rough paths × WGL Institute: WIAS ×

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Now showing 1 - 6 of 6
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    A rough path perspective on renormalization
    (Amsterdam [u.a.] : Elsevier, 2019) Bruned, Y.; Chevyrev, I.; Friz, P.K.; Preiß, R.
    We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned–Hairer–Zambotti (2016), the links with which are discussed in detail. © 2019 The Author(s)
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    Eikonal equations and pathwise solutions to fully non-linear SPDEs
    (New York, NY : Springer, 2016) Friz, Peter K.; Gassiat, Paul; Lions, Pierre-Louis; Souganidis, Panagiotis E.
    We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.
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    Pathwise stability of likelihood estimators for diffusions via rough paths
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Diehl, Joscha; Friz, Peter K.; Mai, Hilmar
    We consider the estimation problem of an unknown drift parameter within classes of non-degenerate diffusion processes. The Maximum Likelihood Estimator (MLE) is analyzed with regard to its pathwise stability properties and robustness towards misspecification in volatility and even the very nature of noise. We construct a version of the estimator based on rough integrals (in the sense of T. Lyons) and present strong evidence that this construction resolves a number of stability issues inherent to the standard MLEs.
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    Solving linear parabolic rough partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Belomestny, Denis; Redmann, Martin; Riedel, Sebastian; Schoenmakers, John
    We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity with 1=3 < 1=2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.
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    The geometry of the space of branched rough paths
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Tapia, Nikolas; Zambotti, Lorenzo
    We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.
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    Rough nonlocal diffusions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Coghi, Michele; Nilssen, Torstein
    We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.