Abstract (EN)

We establish the functional convex order results for two scaled 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) by{dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,X0∈Lp(P),dYt=b(t,Yt,νt)dt+θ(t,Yt,νt)dBt,Y0∈Lp(P),where p≥2, for every t∈[0,T], μt, νt denote the probability distribution of Xt, Yt respectively and the drift coefficient b(t,x,μ) is affine in x (scaled). If we make the convexity and monotony assumption (only) on σ and if σ⪯θ with respect to the partial matrix order, the convex order for the initial random variable X0⪯cvY0 can be propagated to the whole path of process X and Y. That is, if we consider a convex functional F defined on the path space with polynomial growth, we have EF(X)≤EF(Y); for a 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 θ⪯σ and Y0≤X0 with respect to the convex order, 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 principles and the convergence of the Euler scheme of the McKean-Vlasov equation.