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Monotone convex order for the McKean-Vlasov processes

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorLiu, Yating
hal.structure.identifierLaboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
dc.contributor.authorPagès, Gilles
dc.date.accessioned2021-12-09T10:12:02Z
dc.date.available2021-12-09T10:12:02Z
dc.date.issued2022
dc.identifier.issn0304-4149
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22346
dc.language.isoenen
dc.subjectConvex order
dc.subjectMonotone convex order
dc.subjectMcKean-Vlasov process
dc.subjectTruncated Eulerscheme
dc.subject.ddc519en
dc.titleMonotone convex order for the McKean-Vlasov processes
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenIn this paper, we establish the monotone convex order between two R-valued McKean-Vlasov processes X=(Xt)t∈[0,T] and Y=(Yt)t∈[0,T] defined on a filtered probability space (Ω,F,(Ft)t≥0,P) bydXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,X0∈Lp(P)withp≥2,dYt=β(t,Yt,νt)dt+θ(t,Yt,νt)dBt,Y0∈Lp(P),where ∀t∈[0,T],μt=P∘X−1t,νt=P∘Y−1t. If we make the convexity and monotony assumption (only) on b and |σ| and if b≤β and |σ|≤|θ|, then the monotone convex order for the initial random variable X0⪯mcvY0 can be propagated to the whole path of processes X and Y. That is, if we consider a non-decreasing convex functional F defined on the path space with polynomial growth, we have EF(X)≤EF(Y); for a non-decreasing convex functional G defined on the product space involving the path space and its marginal distribution space, we have EG(X,(μt)t∈[0,T])≤EG(Y,(νt)t∈[0,T]) under appropriate conditions. The symmetric setting is also valid, that is, if Y0⪯mcvX0 and |θ|≤|σ|, then EF(Y)≤EF(X) and EG(Y,(νt)t∈[0,T])≤EG(X,(μt)t∈[0,T]). The proof is based on several forward and backward dynamic programming principle and the convergence of the truncated Euler scheme of the McKean-Vlasov equation.
dc.publisher.cityParis
dc.relation.isversionofjnlnameStochastic Processes and their Applications
dc.relation.isversionofjnlvol152
dc.relation.isversionofjnldate2022
dc.relation.isversionofjnlpages312-338
dc.relation.isversionofdoi10.1016/j.spa.2022.06.003
dc.relation.isversionofjnlpublisherElsevier
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.description.ssrncandidatenon
dc.description.halcandidatenon
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2023-02-06T11:45:35Z
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