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- ItemRegularity of SLE in (t,κ) and refined GRR estimates(Berlin ; Heidelberg ; New York, NY : Springer, 2021) Friz, Peter K.; Tran, Huy; Yuan, YizhengSchramm-Loewner evolution ( SLEκ ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by κ times Brownian motion. This yields a (half-plane) valued random field γ=γ(t,κ;ω) . (Hölder) regularity of in γ(·,κ;ω ), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883-924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3-4):413-433, 2014) showed a.s. Hölder continuity of this random field for κ<8(2-3) . In this paper, we improve their result to joint Hölder continuity up to κ<8/3 . Moreover, we show that the SLE κ trace γ(·,κ) (as a continuous path) is stochastically continuous in κ at all κ≠8 . Our proofs rely on a novel variation of the Garsia-Rodemich-Rumsey inequality, which is of independent interest.
- ItemShort-dated smile under rough volatility: asymptotics and numerics(London : Taylor & Francis, 2021) Friz, Peter K.; Gassiat, Paul; Pigato, PaoloIn 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.
- ItemSuperdiffusive limits for deterministic fast-slow dynamical systems(Berlin ; Heidelberg ; New York, NY : Springer, 2020) Chevyrev, Ilya; Friz, Peter K.; Korepanov, Alexey; Melbourne, Ian[ For Abstract, see PDF]
- ItemSingular paths spaces and applications(Philadelphia, Pa. : Taylor & Francis, 2021) Bellingeri, Carlo; Friz, Peter K.; Gerencsér, MátéMotivated 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.
- ItemPrecise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM(Amsterdam [u.a.] : Elsevier, 2022) Friz, Peter K.; Klose, TomWe implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.