A Course on Rough Paths

By Peter K. Friz, Martin Hairer. Universitext, 2020

601 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/74ea82490f0b3f9fd697014a132ec97b26e769ba

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Citations in the corpus

HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “5 of [11] shows that Z “ Rpζq is Hölder rough if ζ stands for space white noise in the flat torus, with L its associated Laplace operator.”
A CLASS OF GROWTH MODELS RESCALING TO KPZ (2015, background): “de, etc. until the expression stabilises and only components in T¯ change from one step to the next. In the simpler context of the KPZ equation, this is explained in the proof of Proposition 15.12 of [FH14]. The abstract derivative of His therefore given by DH= Ψ+I ′(Fˆ) +h·1 +I ′(Fˆ ′I′(Fˆ)) +hI (F”
A regularity structure for rough volatility (2017, methodology): “In the theory of rough paths, one usually takes 𝜑(𝑡) = 𝟏[𝑠,𝑡′](𝑡) for 𝑠 𝑡′ (see also P. K. Friz & Hairer, 2014, Section 13.3.2), which explains why issues of this kind never arise in the rough path framework.”
Averaging dynamics driven by fractional Brownian motion (2019, methodology): “be suﬃcient for our needs. Note that a deterministic version of the sewing lemma was given in [Gub04] and was instrumental for the reformulation of rough path theory [Lyo98] as exposed for example in [FH14]. A multidimensional analogue to the sewing lemma is given by the reconstruction theorem from ”
Noiseless regularisation by noise (2020, methodology): “ sequence of partitions n= f0 = t 0 <t 1 <:::<t k n = tg with mesh j nj= sup i=1;:::;k n jt i t i 1j!0 as n!1, it holds (I) t = lim n!1 kX n 1 i=0 t i;t i+1 : For a proof, see Lemma 4.2 from [26]. Let us point out that estimate (91) is extremely useful even in cases even when I is already k”
Stochastic Normalizing Flows (2020, methodology): “Furthermore, by Friz & Hairer (2014, Theorem 8.5), for any < γ < β , there exist constants C Φ γ and C Ψ γ such that for any β -Hölder continuous rough paths X , Y and ξ, We deduce the following for any t ∈ [0 , T ] and x ∈ R d : (i) (cid:107) Z ( n ) − Z (cid:107) γ → 0 by (18); (ii) using (i) and…”
The infinitesimal generator of the stochastic Burgers equation (2018, background): “In recent years, new solution theories like regularity structures [20,40] or paracontrolled distributions [26,33] were developed for singular SPDEs, see [38] for an up-to-date and fairly exhaustive review.”
Averaging principle for fast-slow system driven by mixed fractional Brownian rough path (2020, methodology): “It is worth mentioning that Hairer and Li [14] considered a slow-fast system where the slow component is driven by fBm and proved the convergence to the averaged solution takes place in probability.”
Short-time near-the-money skew in rough fractional volatility models (2017, background): “A particularly convenient one, that requires no growth restriction on σ , uses continuity of stochastic integration with respect to the rough path (B, Z, ∫ B dZ) = (B, Z, ∫ B̂ dZ) in suitable metrics, for which a LDP is known (Friz and Hairer 2014, Ch 9.3).”
Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity (2018, background): “1.2 that p-variation can be inﬁnite for S∈ Vp(π). When p= 2 it reduces to an ordinary (left) Riemann sum. For p>2 such compensated Riemann sums appear in the construction of ‘rough path integrals’ [17, 19]. Let X∈ Cα([0,T],R) be α-H¨older continuous for some α∈ (0,1), and write q= ⌊α−1⌋. We can e”
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (2018, methodology): “On the one hand, the theory of regularity structures [41, 27, 42] has since grown into a complete framework [13, 19, 12, 44] which can deal with a large class of parabolic equations (in the socalled subcritical regime) such as the dynamical sine-Gordon model [47, 20], the generalized KPZ equation us”