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I am thinking about the martingale version of Monge-Kantorovich Problem.

Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\mathbb{R}^2$ with marginal distributions $\mu$ and $\nu$. Then for a cost function $c(x,y)$, set

$$L(f)=\int_{\mathbb{R}^2}c(x,y)f(x,y)dxdy,\ \ f\in M(\mu,\nu)$$

Then the MKP can be formulated as

$$\sup_{f\in M(\mu,\nu)}L(f)$$

Then we study its martingale version, define

$$\mathcal{M}(\mu,\nu)=\{f\in M(\mu,\nu): \frac{\int_{\mathbb{R}}yf(x,y)dy}{\int_{\mathbb{R}}f(x,y)dy}=x,\ \ \forall x\in\mathbb{R}\}$$

I would like to estimate for $c(x,y)=|x-y|^p$, $p\geq 2$

$$\sup_{f\in\mathcal{M}(\mu,\nu)}L(f)$$

In other words, I look for some function $K(\mu,\nu)$ such that

$$\sup_{f\in\mathcal{M}(\mu,\nu)}L(f)\leq K(\mu,\nu)$$

Does someone have an idea? Many thanks!

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  • $\begingroup$ Did you look at Nizar Touzi's work? $\endgroup$ Oct 8, 2013 at 12:43
  • $\begingroup$ Yes, I am his new phd student. I'd like to prove the compacity of a set of martingale measures, which needs this estimation. $\endgroup$
    – CodeGolf
    Oct 8, 2013 at 12:52
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    $\begingroup$ You should probably just ask Nizar... $\endgroup$ Oct 8, 2013 at 18:08

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