Canonical RDEs and general semimartingales as rough paths

Chevyrev, Ilya; Friz, Peter K.

doi:10.1214/18-aop1264

Cited by 46 publications

(110 citation statements)

References 48 publications

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“…Indeed, it turns out that Marcus' main result and its generalisations (see for precise statements covering the general case given below, and for a rough path perspective) show that for SDEs driven by general semimartingales, the regularised solutions

X_{t}^{ε}

tend to the solution of the following canonical extension: Definition Let

φ (g, x)

denote the value

y (1)

of the ODE

\frac{d y}{d t} = g (y false(t false)), y false(0 false) = x;

then the Marcus canonical extension is the SDE

\begin{matrix} true \\ right X_{t} & = & left X_{0} + \int_{0}^{t} a (X_{s}) d s + \int_{0}^{t} b (X_{s -}) d Z_{s} + \frac{1}{2} \int_{0}^{t} (b ▹ b) (X_{s}) d {false[Z false]}_{s}^{c} \\ left + \sum_{s ⩽ t} ({} φ false(b normalΔ Z_{s}, X_{s -} false) - X_{s -} - b false(X_{s -} false) Δ Z_{s}) . \end{matrix}

…”

Section: Marcus Canonical Extensionmentioning

confidence: 99%

“…Indeed, it turns out that Marcus' main result [37] and its generalisations (see [32] for precise statements covering the general case given below, and [13] for a rough path perspective) show that for SDEs driven by general semimartingales, the regularised solutions X t tend to the solution of the following canonical extension: then the Marcus canonical extension is the SDE…”

Section: Marcus Canonical Extensionmentioning

confidence: 99%

See 1 more Smart Citation

Quasi‐shuffle algebras and renormalisation of rough differential equations

Bruned

Curry

Ebrahimi‐Fard

2019

Bull. London Math. Soc.

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The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of B‐series. For this purpose, we present a so‐called arborification of the Hoffman–Ihara theory of quasi‐shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi‐shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.

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X_{t}^{ε}

tend to the solution of the following canonical extension: Definition Let

φ (g, x)

denote the value

y (1)

of the ODE

\frac{d y}{d t} = g (y false(t false)), y false(0 false) = x;

then the Marcus canonical extension is the SDE

\begin{matrix} true \\ right X_{t} & = & left X_{0} + \int_{0}^{t} a (X_{s}) d s + \int_{0}^{t} b (X_{s -}) d Z_{s} + \frac{1}{2} \int_{0}^{t} (b ▹ b) (X_{s}) d {false[Z false]}_{s}^{c} \\ left + \sum_{s ⩽ t} ({} φ false(b normalΔ Z_{s}, X_{s -} false) - X_{s -} - b false(X_{s -} false) Δ Z_{s}) . \end{matrix}

…”

Section: Marcus Canonical Extensionmentioning

confidence: 99%

Section: Marcus Canonical Extensionmentioning

confidence: 99%

Quasi‐shuffle algebras and renormalisation of rough differential equations

Bruned

Curry

Ebrahimi‐Fard

2019

Bull. London Math. Soc.

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show abstract

“…These new developments significantly generalize an earlier work by Williams [Wil01]. While [Wil01] already provides a pathwise meaning to stochastic differential equations driven by certain Lévy processes, [FS17,CF17] develop a more complete picture about càdlàg rough paths, including rough path integration, differential equations driven by càdlàg rough paths and the continuity of the corresponding solution maps. We refer to [LCL07,FV10b,FH14] for detailed introductions to classical rough path theory.…”

Section: Introductionmentioning

confidence: 61%

“…Very recently, the notion of càdlàg rough paths was introduced by Friz and Shekhar [FS17] (see also [CF17,Che17]) extending the well-known theory of continuous rough paths initiated by Lyons [Lyo98]. These new developments significantly generalize an earlier work by Williams [Wil01].…”

Section: Introductionmentioning

confidence: 94%

Examples of Itô càdlàg rough paths

Liu

Prömel

2018

Proc. Amer. Math. Soc.

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“…Also, there are some very recent works, see e.g. Chevyrev and Friz [CF17], where rough differential equations are studied in the spirit of Marcus canonical stochastic differential equations by dropping the assumption of continuity prevalent in the rough path literature. Therefore, we hope that Gubinelli's [Gub04] approach of Lyons' theory of integration over rough paths may be integrated with [CF17] and our approach to gain newer insight into the analysis of constrained SPDEs.…”

mentioning

confidence: 99%

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

Brzeźniak

Hornung

Manna

2019

Stoch PDE: Anal Comp

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We construct a martingale solution of the stochastic nonlinear Schrödinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in H 1 on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of càdlàg functions and Jakubowski's generalization of the Skorohod-Theorem to nonmetric spaces.

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Canonical RDEs and general semimartingales as rough paths

Cited by 46 publications

References 48 publications

Quasi‐shuffle algebras and renormalisation of rough differential equations

Quasi‐shuffle algebras and renormalisation of rough differential equations

Examples of Itô càdlàg rough paths

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

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