Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

By Peter K. Friz, Nicolas Victoir. 2010

621 citations recorded by the source corpus.

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Citations in the corpus

A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “rovide robust analysis of ODEs and a novel way to deﬁne solutions of stochastic diﬀerential equations driven by non semimartingale signals. For a comprehensive review see the book of Friz and Victoir [28] and the lecture notes of Lyons, Caruana and Lévy [57] or the more recents ones of Friz and Hair”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “First, we need a standing assumption that regards the regularity of R(s, t) (expressed in terms of its so called 2D ρ-variation, see [17]) and complementary Young regularity of X and its Cameron–Martin space.”
Heat semigroup and singular PDEs (2015, methodology): “1 + log (τ + r τ ) ) 1 2 dτ 2p dr r , where we have used Gaussian hypercontractivity; see [15].”
Smooth approximation of stochastic differential equations (2014, methodology): “n on bcan be relaxed to C2+. 34 D. KELLY AND I. MELBOURNE This result has been used in several contexts [7, 25], so we only sketch the proof. Proof of Theorem 9.1. First, suppose that a∈C1+, b∈C3. By [17], Theorem 12.10, we know that the map (Un,Un) → Xn is continuous with respect to the ργ topology”
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems (2013, methodology): “tial diﬀerential equations (SPDEs) beyond the Brownian motion case. One of these attempts is given by the Rough Path Theory, and we refer to the monographs by Lyons and Qian [16] and Friz and Victoir [5] for a comprehensive presentation of this theory. When the driving process is a Ho¨lder continuou”
Global Dynamics for the Two-dimensional Stochastic Nonlinear Wave Equations (2020, unknown): “) ζ(t−s) α˙ ≤ max(Ft1,t2,e) (2.2) for any 0 ≤ t1 ≤ t2 ≤ T with t2 −t1 ≤ 1, where ζ(t) is deﬁned by ζ(t) = ˆt 0 τθ−1 ˆ log 1 + 4 τ2 ˙1 α dτ. (2.3) When α = 2, Lemma 2.2 reduces to Corollary A.5 in [16]. While Lemma 2.2 for general α > 0 follows in an analogous manner, we present a proof for reade”
Stochastic Calculus with Respect to Gaussian Processes (2014, background): “The reader interested in this approach, that also provides Itô formulas, will find in [13] a very complete overview.”
Discretely sampled signals and the rough Hoff process (2013, background): “3 of [11], it follows that almost surely z = π(Q,W ;L) (0, (0, y0); (X, ψ)) .”
Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions (2014, result): “There are also a number of results on the rate of convergence of Euler-type numerical schemes in this case; see, for instance, the paper by Deya, Neuenkirch and Tindel [4] for a Milstein-type scheme without Lévy area in the case 13 < H < 1 2 , the paper by Friz and Riedel [5] for the N -step Euler s”
Canonical RDEs and general semimartingales as rough paths (2017, methodology): “(iv) Recall by Wiener’s characterization [20], Theorem 8.”
Pathwise stochastic integrals for model free finance (2013, methodology): “have an associated rough path of ﬁnite 2-variation, up to logarithmic corrections. For such rough 32 N. Perkowski and D.J. Pro¨mel paths, the integral extends to integrands F∈ C1, see Chapter 10.5 of [20]. For typical price paths (but not for the area), it is shown in [40], Section 4.3, that they ar”
Rough path limits of the Wong–Zakai type with a modified drift term (2009, methodology): “m on the r.h.s goes to 0 as |D|→0 and this ﬁnishes the proof. 4.2. Optimality of RDE estimates. At last, we use theorem 2 to establish optimality of two important RDE estimates (proved in [Lyo98] and [FV08b]) for the case of paths with p-variation, p ∈N. The second part of the following theorem sett”