Monotone convex order for the McKean–Vlasov processes

This paper is a continuation of our previous paper (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] 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)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 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 principles and the convergence of the truncated Euler scheme of the McKean–Vlasov equation.