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Author: Friz, Peter K. × Subject: Rough paths ×

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Now showing 1 - 4 of 4
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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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    Singular paths spaces and applications
    (Philadelphia, Pa. : Taylor & Francis, 2021) Bellingeri, Carlo; Friz, Peter K.; Gerencsér, Máté
    Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modeled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting, this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime κ<1. We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalization methods.
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    Short-dated smile under rough volatility: asymptotics and numerics
    (London : Taylor & Francis, 2021) Friz, Peter K.; Gassiat, Paul; Pigato, Paolo
    In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small-noise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782–832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence.
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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.