dc.contributor.author
Matsuda, Toyomu
dc.date.accessioned
2023-10-16T09:00:26Z
dc.date.available
2023-10-16T09:00:26Z
dc.identifier.uri
https://refubium.fu-berlin.de/handle/fub188/40864
dc.identifier.uri
http://dx.doi.org/10.17169/refubium-40585
dc.description.abstract
The thesis explores stochastic calculus for fractional Brownian motion. Our approach builds upon a novel technique called stochastic sewing, originally introduced by Lê [Electron. J. Probab. 25:1-55, 2020]. The stochastic sewing has been effectively used to obtain sharp estimates on stochastic Riemann sums.
The main result of the thesis is an extension of Lê’s stochastic sewing, which we refer to as the shifted stochastic sewing. This extension takes advantage of asymptotic decorrelation in stochastic Riemann sums and can be seen as a combination of Lê’s stochastic sewing and the asymptotic independence formulated by Picard [Ann. Probab. 36(6): 2235-2279, 2008].
As applications of the shifted stochastic sewing, we address two important problems in fractional stochastic calculus. Firstly, we characterize the local time of the fractional Brownian motion via level crossings, extending the classical work of Lévy to the fractional setting. Secondly, we establish the pathwise uniqueness of Young and rough differential equations driven by fractional Brownian motion. This result optimizes the regularity of the noise coefficient, which is consistent with the Brownian setting.
Additionally, we demonstrate strong regularization by fractional noise for differential equations with integrable drifts. This result can be viewed as a fractional analogue of the celebrated work by Krylov and Röckner [Probab. Theory Relat. Fields 131: 154–196, 2005].
en
dc.format.extent
x, 189 Seiten
dc.rights.uri
http://www.fu-berlin.de/sites/refubium/rechtliches/Nutzungsbedingungen
dc.subject
Fractional stochastic calculus
en
dc.subject
Fractional Brownian motion
en
dc.subject
Stochastic sewing
en
dc.subject.ddc
500 Natural sciences and mathematics::510 Mathematics::519 Probabilities and applied mathematics
dc.title
Fractional Stochastic Calculus via Stochastic Sewing
dc.contributor.gender
male
dc.contributor.firstReferee
Perkowski, Nicolas
dc.contributor.furtherReferee
Nourdin, Ivan
dc.date.accepted
2023-09-12
dc.identifier.urn
urn:nbn:de:kobv:188-refubium-40864-6
refubium.affiliation
Mathematik und Informatik
dcterms.accessRights.dnb
free
dcterms.accessRights.openaire
open access