Functional convex order for the scaled McKean-Vlasov processes
Résumé
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 ∈ L p (P), dYt = b(t, Yt , νt)dt + θ(t, Yt , νt)dBt, Y0 ∈ L p (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 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, (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 E F (Y) ≤ E F (X) and E G Y, (νt) 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.
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