Let X=(X)0≤t≤1 be a continuous martingale starting at 0, then denote by μ and ν the probability laws of ∫10Xtdt and X1. Then it is easy to see that the couple (μ,ν) is increasing in convex order, i.e.
∫Rf(x)dμ(x) ≤ ∫Rf(x)dν(x)
holds for all convex functions f:R→R of linear growth, see also “Peacocks and Associated Martingales, with Explicit Constructions” for further details. Now my question is given by probability measures μ and ν on R, could we give some conditions on the couple (μ,ν) such that there exists a continuous martingale X=(X)0≤t≤1 satisfying
X1∼μ and ∫10Xtdt∼ν?