Functional convex order for the scaled McKean-Vlasov processes
Abstract
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 (, F, (Ft)t≥0, P) by \[cases dXt= b(t, Xt, μt)dt+σ(t, Xt, μt)dBt, \;\;X0∈ Lp(P),\\ dYt\,= b(t, \,Yt\,,\, t)dt+θ(t, \,Yt\,,\, t)dBt, \;\;Y0∈ Lp(P), cases\] where p≥2, for every t∈[0, T], μt, t 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 σθ with respect to the partial matrix order, the convex order for the initial random variable X0 \,cv Y0 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 E\,G(X, (μt)t∈[0, T])≤ E\,G(Y, (t)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 E\,F(Y) ≤ E\,F(X) and E\,G(Y, (t)t∈[0, T])≤ E\,G(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.