Monotone convex order for the McKean–Vlasov processes

This paper is a continuation of our previous (Liu and Pagès, 2020). In this paper, we establish the monotone convex order (see further (1.1)) between two R-valued McKean–Vlasov processes X=(Xt)t∈[0,T] Y=(Yt)t∈[0,T] defined on filtered probability space (Ω,F,(Ft)t≥0,P) by dXt=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∘Xt−1,νt=P∘Yt−1.If make convexity monotony assumption (only) b |σ| if b≤β |σ|≤|θ|, then for initial random variable X0⪯mcvY0 can be propagated to whole path X Y. That is, consider non-decreasing functional F with polynomial growth, have EF(X)≤EF(Y); G product involving its marginal distribution space, EG(X,(μt)t∈[0,T])≤EG(Y,(νt)t∈[0,T]) under appropriate conditions. The symmetric setting also valid, that Y0⪯mcvX0 |θ|≤|σ|, EF(Y)≤EF(X) EG(Y,(νt)t∈[0,T])≤EG(X,(μt)t∈[0,T]). proof based several forward backward dynamic programming principles convergence truncated Euler scheme equation.