Sensitivity of rough differential equations: An approach through the Omega lemma

Coutin, Laure; Lejay, Antoine

doi:10.1016/j.jde.2017.11.031

Cited by 10 publications

(12 citation statements)

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“…We noted in [21,Remark 1.4] that every delayed controlled path based on X can be seen as a usual controlled path based on (X, X •−r ) and vice versa. Using this identification, the assertion just follows from [12,Proposition 5].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…More precisely, we will give sufficient conditions under which this map is differentiable in the initial condition, which means differentiability in Fréchet-sense on the space of controlled paths. To prove our result, we will follow a similar strategy as in [3] and [12]. Definition 2.2.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…This shows that we can use the implicit function theorem on Banach spaces (cf. [1, 2.5.7 Implicit Function Theorem] or [12,Theorem 1]) to see that there is a neighbourhood U aroundξ such that for every ξ ∈ U , there are functions (z 1,ξ s ) 0 s τ3 with the property that Λ…”

Section: Preliminaries and Notationmentioning

confidence: 99%

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A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Ghani¹,

Riedel²

2021

DCDS-B

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Section: Preliminaries and Notationmentioning

confidence: 99%

Section: Preliminaries and Notationmentioning

confidence: 99%

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A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Ghani¹,

Riedel²

2021

DCDS-B

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“…Classical results from rough paths theory ensure that ψ is a continuous function from C 1 pr0, T s, R d q with values in the space C α pr0, T s, R d q -see e.g. Theorem 3 of [10]. This is where we need the assumption that G is p2`γq-Hölder, rather than just p1`γq-Hölder, as in the proof of Theorem 1.4 given in the next section.…”

Section: -Upper Semicontinuous Driftmentioning

confidence: 99%

Young and rough differential inclusions

2020

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We define in this work a notion of Young differential inclusion dz t P F pz t qdx t , for an α-Hölder control x, with α ą 1{2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Hölder continuous set-valued map on the interval r0, 1s has a selection with finite p-variation, for p ą 1{γ. We also give a notion of solution to the rough differential inclusion dz t P F pz t qdt`Gpz t qdX t , for an α-Hölder rough path X with α P`1 3 , 1 2 ‰ , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values. 1-Introduction 1.1-Setting One of the motivations for considering differential inclusions comes from the study of differential equations with discontinuous coefficients. In the setting of a possibly timedependent widely discontinuous vector field on R d , it makes sense to replace the dynamical prescription 9 z t " f pt, z t q, by 9 z t P F pt, z t q, where F pt, zq is here the closed set of cluster points ot F ps, wq, as ps, wq converges to pt, zq. This somehow accounts for the impossiblity to make measurements with absolute precision. It also makes sense to take for F pt, zq the convex hull of the former set. A set-valued application F is a map from r0, T sˆR d into the power set of R d. Different natural setvalued extensions of f may have different regularity properties; in any case, switching from the differential equation prescription to the differential inclusion formulation somehow allows to account for the uncertainty in the modeling. A Caratheodory solution of the differential inclusion 9 z t P F pt, z t q, t P r0, T s, z 0 " ξ P R d (1.1) is an absolutely continuous path z started from ξ, whose derivative 9 z satisfies 9 z t P F pt, z t q

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“…Remark 2. 3 We stress that the trace-class condition is the only requirement we impose on the covariance of the fractional Brownian motion in contrast to [19].…”

Section: Introductionmentioning

confidence: 99%

Local mild solutions for rough stochastic partial differential equations

Hesse

Neamţu

2019

Journal of Differential Equations

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We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space. We are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. We thank the referee for carefully reading the manuscript and for the valuable suggestions.AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1). 1 index H ∈ (1/3, 1/2]. In order to solve (1.1) we need to give a meaning of the rough integral t 0 S(t − r)G(y r )dω r . (1.2) Results in this context are available in [10] via fractional calculus and in Gubinelli et al [11], [4] [12], [13] using rough paths techniques. In this work, we combine Gubinelli's approach with the arguments employed by [10] to solve (1.1). This theory should hopefully be more simple in order to investigate the long-time behavior of such equations using a random dynamical systems approach such in [1], [8] or [2]. In this work we establish only the existence of a local mild solution. We investigate in a forthcoming paper global solutions and random dynamical systems for (1.1) as in [8]. This work should be seen as a first step in order to close the gap between rough paths and random dynamical systems in infinite-dimensional spaces. Since the fractional Brownian motion is not a semi-martingale, the construction of an appropriate integral represents a challenging problem. This has been intensively investigated and numerous results and various techniques are available, see [26], [8], [25], [16], [19] and the references specified therein. There is a huge literature where certain tools from fractional calculus (i.e. fractional/compensated fractional derivative/integral) are employed to give a pathwise meaning of the stochastic integral with respect to the fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) or H ∈ (1/3, 1/2]. A different method which has been recently introduced and explored is given by the rough path approach of Gubinelli et. al. [11], [13], [12]. This goes through if H ≤ 1/2. Moreover it is suitable to define (1.2) not only with respect to the fractional Brownian motion but also to Gaussian processes for which the covariance function satisfies certain structure, see [6] or [5, Chapter 10]. An overview on the connection between rough paths and fractional calculus can be looked up in [16]. Of course, in some situations, various other techniques for H ≤ 1/3 are available. After using an appropriate integration theory with respect to the fractional Brownian motion, the next step is to analyze SDEs/SPDEs driven by this kind of noise. There is a growing interest in establishing suitable properties of the solution under several assumptions on the coefficients, consult [19], [20], [14], [7], [8], [26] and the references specified therein. To our aims we ...

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Sensitivity of rough differential equations: An approach through the Omega lemma

Cited by 10 publications

References 50 publications

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Young and rough differential inclusions

Local mild solutions for rough stochastic partial differential equations

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