A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations

Abstract

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...

Existence, uniqueness and stability of semi-linear rough partial differential equations

“…A step towards this goal is to consider the diffusion X as a solution to the rough differential equation [4]. We will denote by Φ the flow generated by (3.3).…”

Backward stochastic differential equations with Young drift

“…The proofs are based on a transformation of the SPDE into a PDE with random coefficients and a study of the latter using PDE methods. In (Diehl et al 2015), the convergence of solutions corresponding to smooth approximations of η is shown using a linear Feynman-Kac formula. In (Diehl et al 2014), these results are extended to show that the limit actually solves an integral equation.…”

“…In (Diehl et al 2015), well-posedness of the corresponding mixed SDE is established by first constructing a joint rough path "above" W and η. The deterministic theory of rough paths then allows mixed SDEs to be solved.…”

Scalar conservation laws with rough flux and stochastic forcing

“…Let us now justify (5.16). It was shown in [14,Theorem 8] that the RDE solution to the characteristic system (5.5) corresponding to the joint lift Λ constructed in (2.1) can be obtained as limit of SDE solutions to dϕ n,0…”