Examples of Renormalized SDEs

Bruned, Yvain; Chevyrev, Ilya; Friz, Peter K.

doi:10.1007/978-3-319-74929-7_19

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…Moreover, v ε can be chosen so that the renormalized limit is indeed B Strat . Details of this second-level renormalization can be found in [BCF17]. We also point out that higher order renormalization can be expected in the presence of highly oscillatory fields, which also points to some natural connections with homogenization theory.…”

Section: Higher-order Translation and Renormalization In Finite-dimenmentioning

confidence: 66%

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A rough path perspective on renormalization

Bruned

Chevyrev

Friz

et al. 2019

Journal of Functional Analysis

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We revisit (higher-order) translation operators on rough paths, in both the geometric and branched setting. As in Hairer's work on the renormalization of singular SPDEs we propose a purely algebraic view on the matter. Recent advances in the theory of regularity structures, especially the Hopf algebraic interplay of positive and negative renormalization of Bruned-Hairer-Zambotti (2016), are seen to have precise counterparts in the rough path context, even with a similar formalism (short of polynomial decorations and colourings). Renormalization is then seen to correspond precisely to (higher-order) rough path translation.

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Section: Higher-order Translation and Renormalization In Finite-dimenmentioning

confidence: 66%

“…Remark 33. In [BCF17], two examples are studied of families of random bounded variation paths (X ε ) ε>0 whose canonical lifts to geometric rough paths (X ε ) ε>0 diverge as ε → 0. In particular, ODEs driven by X ε in general also fail to converge.…”

Section: Rough Differential Equations 51 Translated Rough Paths Are mentioning

confidence: 99%

A rough path perspective on renormalization

Bruned

Chevyrev

Friz

et al. 2019

Journal of Functional Analysis

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“…The same question about rough paths has been asked recently in , and indeed it could have been asked much earlier. Maybe this has not happened because the motivation was less compelling; although one can construct examples of rough paths depending on a positive parameter which do not converge as the parameter tends to 0, this phenomenon is the exception rather than the rule.…”

Section: Introductionmentioning

confidence: 93%

“…Observe that since

ψ

is a Hopf algebra morphism, in particular a coalgebra morphism, then

false(ψ \otimes ψ false) {normalΔ}^{'} τ = {\overset{Δ}{true}}^{'} ψ false(τ false) = {\overset{Δ}{true}}^{'} ψ_{n - 1} false(τ false), τ \in {scriptT}_{n},

since trees are primitive elements in

T (T)

, being single‐letter words. From the proof of [, Lemma 4.9] we are able to see that in fact

ψ_{n - 1}

is given by the recursion

ψ_{n - 1} = m_{\otimes} false(ψ \otimes prefixid false) {normalΔ}^{'}

on the linear span of

T_{n}

(see also [, Definition 1, Section 6]). Example Here are some examples of the action of

ψ

on some trees:

…”

Section: The Hairer–kelly Constructionmentioning

confidence: 99%

The geometry of the space of branched rough paths

Tapia

Zambotti

2020

Proc. London Math. Soc.

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We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between these two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker–Campbell–Hausdorff formula, on a constructive version of the Lyons–Victoir extension theorem and on the Hairer–Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.

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“…Thus, BPHZ renormalization is key in the theory of regularity structures. Also, study of the relation between rough paths and regularity structures gave rise to discussion on the renormalization of rough paths, given that they are a counterpart of regularity structures ( [3], [4], [5]).…”

Section: Introductionmentioning

confidence: 99%

BPHZ Renormalization in Gaussian Rough Paths

Ito

2021

Preprint

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We construct a procedure for Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization of a rough path in view of the relation between rough path theory and regularity structure. We also provide a plain expression of the BPHZ-renormalized model in a rough path. BPHZ renormalization plays a central role in the theory of singular stochastic partial differential equations and assures the convergence of the model in the regularity structure. Here we demonstrate that the renormalization is also effective in a rough path setting by considering its application to rough path theory and mathematical finance.

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Examples of Renormalized SDEs

Cited by 5 publications

References 12 publications

A rough path perspective on renormalization

A rough path perspective on renormalization

The geometry of the space of branched rough paths

BPHZ Renormalization in Gaussian Rough Paths

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