FUNCTIONAL CONVEX ORDER FOR THE SCALED MCKEAN-VLASOV PROCESSES

We establish the functional convex order results for two scaled McKean- Vlasov processes X = (X-t)(t is an element of[0 ,T] )and Y = (Y-t)(t is an element of[0 ,T]) defined on a filtered probability space (Omega,F,(F-t)(t >= 0) , P) by{dX(t )= b(t, X-t , mu(t))d(t) + sigma (t, X-t , mu(t))dB(t) , X-0 is an element of L-p(P),dY(t) = b(t,Y-t ,nu(t))d(t) + theta(t, Y-t , nu(t))dB(t) , Y-0 is an element of L-p(P), where p >= 2, for every t is an element of [0 , T ] , mu(t) , nu t denote the probability distribution of Xt , Yt respectively and the drift coefficient b(t, x, mu) is affine in x (scaled). If we make the convexity and monotony assumption (only) on sigma and if sigma less than or similar to theta with respect to the partial matrix order, the convex order for the initial ran-dom variable X-0 less than or similar to(cv) Y-0 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, (mu(t))(t is an element of[0 ,T])) <= EG(Y, (nu(t))(t is an element of[0 ,T])) under appropriate conditions. The symmetric setting is also valid, that is, if theta less than or similar to sigma and Y-0 <= X-0 with respect to the convex order, then EF(Y) <= EF(X) and EG(Y, (nu(t))(t is an element of[0 ,T]))<= EG(X, (mu(t))(t is an element of[0 ,T])).The proof is based on several for-ward and backward dynamic programming principles and the convergence of the Euler scheme of the McKean-Vlasov equation. Two applications of these results, to mean field control and mean field games, are proposed.