27 pagesInternational audienceWe analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems

Abstract. New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or ). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the saving possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.

Under the key assumption of finite ρ-variation, ρ ∈ [1, 2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ = 1 resp. ρ = 1/ (2H), we recover and extend the respective results of [Hu-Nualart; Rough path analysis via fractional calculus; TAMS 361 (2009) 2689-2718] and [Deya-Neuenkirch-Tindel; A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion;

A. We give meaning and study the regularity of di erential equations with a rough path term and a Brownian noise term, that is we are interested in equations of the typewhere η is a deterministic geometric, step-2 rough path and B is a multi-dimensional Brownian motion. En passant, we give a short and direct argument that implies integrability estimates for rough di erential equations with Gaussian driving signals which is of independent interest. IThe contribution of this article is twofold: rstly, we give meaning to di erential equations of the typethat is, for a deterministic, step 2-rough path η we are looking for a stochastic process S η that is adapted to σ (B) and study the regularity of the map η → S η . Secondly, we take this as an opportunity to revisit the integrability estimates of solutions of rough di erential equations driven by Gaussian processes. If either b ≡ 0 or c ≡ 0 then rough path theory [26,27,29,19,18] or standard Itōcalculus allow (under appropriate regularity assumptions on the vector elds (a, b, c)) to give meaning to (1.1). However, in the generic case when the vector elds b and c have a non-trivial Lie bracket, any notion of a solution (that is consistent with an Itō-Stratonovich calculus) must take into account the area swept out between the trajectories of B and η. A natural approach is to identify S η as the RDE solution of2000 Mathematics Subject Classi cation. 60H10,60H30,60H05,60G15,60G17. Key words and phrases. Existence of path integrals, Integrability of rough di erential equations with Gaussian signals, Clark's robustness problem in nonlinear ltering, Viscosity solutions of RPDEs. 1 A LEVY-AREA BETWEEN BROWNIAN MOTION AND ROUGH PATHS WITH APPLICATIONS 2where Λ is a joint, step-2 rough path lift between the enhanced Brownian motion B = 1 + B +´B ⊗ •dB and η, and (r, Λ) is the joint rough path between the random rough path Λ and the bounded variation path r → r. While the existence of a joint lift between a continuous bounded variation path and any rough path is trivial (via integration by parts), the existence of a joint lift between two given step-2 rough paths is more subtle and in general not possible. More precisely, let α ∈ 1 3 , 1 2 and denote with C 0,α R d the space of geometric, step-2, α-Hölder rough paths over R d (we often only write C 0,α and d is chosen according to context). Fix two geometric, step-2 rough paths η = (1 + ηIn general, one cannot hope to nd a joint rough path lift, i.e. a geometric rough path λ = 1 + λ 1 + λ 2 ∈ C 0,α R d+e such that (formally)since the entries on the cross-diagonal of λ 2 are not well-de ned. (What is guaranteed by the extension theorem in [28] is that there exists a weak geometric rough path λ such that λ 1 = η 1 , b 1 , however this λ is highly non-unique and no consistency with η or b on the second level is guaranteed).In Section 2 we show that in the case when the deterministic rough path b is replaced by enhanced Brownian motion B, there does indeed exists a stochastic process Λ which merits in a certain sense to be called...