Show simple item record

Comparison of viscosity solutions of semi-linear path-dependent PDEs

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorRen, Zhenjie
hal.structure.identifierCentre de Mathématiques Appliquées [CMAP]
dc.contributor.authorTouzi, Nizar
hal.structure.identifier
dc.contributor.authorZhang, Jianfeng
dc.date.accessioned2020-01-28T10:36:33Z
dc.date.available2020-01-28T10:36:33Z
dc.date.issued2020
dc.identifier.issn0363-0129
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/20484
dc.language.isoenen
dc.subjectviscosity solutionsen
dc.subjectoptimal stoppingen
dc.subjectstochastic controlen
dc.subjectpath-dependent PDEsen
dc.subject.ddc519en
dc.titleComparison of viscosity solutions of semi-linear path-dependent PDEsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherUniversity of Southern California;United States
dc.description.abstractenThis paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204--236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645--3654; SIAM J. Control Optim., 51 (2013), pp. 4274--4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204--236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.en
dc.relation.isversionofjnlnameSIAM Journal on Control and Optimization
dc.relation.isversionofjnlvol58en
dc.relation.isversionofjnlissue1en
dc.relation.isversionofjnldate2020
dc.relation.isversionofjnlpages277–302en
dc.relation.isversionofdoi10.1137/19M1239404en
dc.contributor.countryeditoruniversityotherUNITED STATES
dc.relation.isversionofjnlpublisherSIAM - Society for Industrial and Applied Mathematicsen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2020-01-28T10:33:01Z
hal.author.functionaut
hal.author.functionaut
hal.author.functionaut


Files in this item

Thumbnail
Name:
RZTvisco-submit(1).pdf
Size:
424.4Kb
Format:
PDF
View/Open

This item appears in the following Collection(s)

Show simple item record