2023/03/02 by David Hobson, Hobson, David, Dominykas Norgilas +1
Mathematics · #60G42 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2303.01578
openalex publication_date 2023/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give an injective martingale coupling; in particular, given measures μ and ν in convex order on \mathbb R such that ν is continuous, we construct a martingale transport such that for each y in the support of the target law ν there is a \em unique x in a support of the initial law μ such that (some of) the mass at x is transported to y. Then π has disintegration π(dx,dy) = ν(dy) δθ(y)(dx) for some function θ. More precisely we construct a martingale coupling π of the measures μ and ν such that there is a set Γμ such that μ(Γμ)=1 and a disintegration (πx)x ∈ Γμ of π of the form π(dx,dy) = πx(dy) μ(dx) such that, with Γπx a support of πx, we have # \ x ∈ Γμ: y ∈ Γπx \ ∈ \ 0,1 \ for all y and \ y : # \ x ∈ Γμ: y ∈ Γπx \ = 1 \ = supp(ν). Moreover, if μ is continuous we may take Γπx = supp(πx) for each x. However, we cannot also insist that Γμ= supp (μ).