Monotone convex order for the McKean-Vlasov processes
Liu, Yating; Pagès, Gilles (2022), Monotone convex order for the McKean-Vlasov processes, Stochastic Processes and their Applications, 152, p. 312-338. 10.1016/j.spa.2022.06.003
TypeArticle accepté pour publication ou publié
Journal nameStochastic Processes and their Applications
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
Abstract (EN)In 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.
Subjects / KeywordsConvex order; Monotone convex order; McKean-Vlasov process; Truncated Eulerscheme
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