Multidimensional Stochastic Processes as Rough Paths 2010
DOI: 10.1017/cbo9780511845079.013
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Abstract: A hybrid theory of rough stochastic analysis is built. It seamlessly combines the advantages of both Itô's stochastic -and Lyons' rough differential equations. Well-posedness of rough stochastic differential equation is obtained, under natural assumptions and with precise estimates; many examples and applications are mentioned. A major role is played by a new stochastic variant of Gubinelli's controlled rough paths spaces, with norms that reflect some generalized stochastic sewing lemma, and which may prove us… Show more

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Cited by 2 publications
(7 citation statements)
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“…Several results of local Lipschitz continuity have been established recently, especially in [6], [9], [16], [17], although not completely satisfactory from a practical point of view. So we decided not to reproduce (and take advantage of) them here.…”
Section: Quantization Of the Sde And Main Resultsmentioning
confidence: 99%
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“…Several results of local Lipschitz continuity have been established recently, especially in [6], [9], [16], [17], although not completely satisfactory from a practical point of view. So we decided not to reproduce (and take advantage of) them here.…”
Section: Quantization Of the Sde And Main Resultsmentioning
confidence: 99%
“…This convergence will hold with respect to distance introduced in the rough path theory (see [25,14,6,9,26]) which always implies convergence with respect to the sup norm. The reason is that our result will appear as an application of (variants of the) the celebrated Universal Limit Theorem originally established by T. Lyons in [25].…”
Section: Introductionmentioning
confidence: 93%
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“…The stochastic sewing lemma introduced in [Lê20] is an extension of the sewing lemma which takes into account stochastic cancellations. Since its introduction, the stochastic sewing lemma has caught some attention and led to new interesting applications; ranging from regularization by noise problem [ABLM20, HP21, Ger20], numerical methods for stochastic differential equations [DGL21,BDG19], rough stochastic differential equations [FHL21] to averaging principle with fractional dynamics [HL20,LS20]. While the sewing lemma is applicable for processes in any Banach spaces, its stochastic version from [Lê20] is only applicable for stochastic processes in R d .…”
Section: Introductionmentioning
confidence: 99%