2021/04/21 by Liu, Yating, Pagès, Gilles
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2104.10421
In this paper, we establish the monotone convex order 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 amp;dXt=b(t, Xt, μt)dt+σ(t, Xt, μt)dBt, X0∈ Lp(ℙ) with p≥ 2,
amp;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\preceq mcv Y0 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 𝔼 F(X)≤ 𝔼 F(Y); for a non-decreasing 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 Y0\preceq mcv X0 and |θ|≤ |σ|, 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 principle and the convergence of the truncated Euler scheme of the McKean-Vlasov equation.