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- ItemThe enhanced Sanov theorem and propagation of chaos(Amsterdam [u.a.] : Elsevier, 2017) Deuschel, Jean-Dominique; Friz, Peter K.; Maurelli, Mario; Slowik, MartinWe establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.
- ItemFrom rough path estimates to multilevel Monte Carlo(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Friz, Peter K.; Riedel, Sebastian; Schoenmakers, John G.M.Discrete approximations to solutions of stochastic differential equations are well-known to converge with strong rate 1=2. Such rates have played a key-role in Giles multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H > 1=4 as special case).
- ItemA regularity structure for rough volatility(Oxford [u.a.] : Wiley-Blackwell, 2019) Bayer, Christian; Friz, Peter K.; Gassiat, Paul; Martin, Jorg; Stemper, BenjaminA new paradigm has emerged recently in financial modeling: rough (stochastic) volatility. First observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, rough volatility captures parsimoniously key-stylized facts of the entire implied volatility surface, including extreme skews (as observed earlier by Alòs et al.) that were thought to be outside the scope of stochastic volatility models. On the mathematical side, Markovianity and, partially, semimartingality are lost. In this paper, we show 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 provide a new and powerful tool to analyze rough volatility models.
- 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.
- ItemEikonal 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.
- ItemStochastic model for LFP-electrodes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dreyer, Wolfgang; Friz, Peter K.; Gajewski, Paul; Guhlke, Clemens; Maurelli, MarioIn the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.
- 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.
- ItemOn the regularity of SLE trace(Cambridge : Cambridge Univ. Press, 2017) Friz, Peter K.; Tran, HuyWe revisit regularity of SLE trace, for all κ≠8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1+κ/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.
- 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.