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I encounter the following problem during the course of my research: Given a random variable $Y=(Y_1,Y_2)$ with values in $\mathbb R^2$ and the cost function $c(x,y)=(x_1-y_1)(x_2-y_2)$ where $x=(x_1,x_2),y=(y_1,y_2)$ are points in $\mathbb R^2$, we consider

$$\min\mathbb E(c(X,Y))$$

where the minimization is taken over all $Y$-measurable random variable $X$ such that $(X,Y)$ forms a 1-step martingale. Equivalently, we would like to minimize $\min\mathbb E(c(T(Y),Y))$ over all measurable map $T:\mathbb R^2\rightarrow \mathbb R^2$, so that $(T(Y),Y)$ is a martingale. (thus this problem is similar to the Monge optimal transport problem in a "backward" sense, plus a martingale constraint).

We know that solution exists if the law of $Y$ is nice(say, absolutely continuous w.r.t. the Lebesgue measure). In general, we expect that solution does not exist.

My question is therefore to construct an example where there is no solution to the above problem.

Any advise and hint are greatly appreciated!

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  • $\begingroup$ What exactly do you mean by saying that $\mathbb R^2$-valued random variables $X$ and $Y$ form a ``1-step martingale''? $\endgroup$
    – R W
    Commented Feb 19, 2018 at 22:07
  • $\begingroup$ I mean $E(Y_i |X)=X_i$ for $i=1,2$. $\endgroup$
    – Ryan
    Commented Feb 19, 2018 at 22:48

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Since the law of $X$ is not fixed, this problem has nothing to do with optimal transport (and thus nothing to do with martingale optimal transport).

This is just a typical optimal control problem, where the distribution of the target is known, but any initial distribution is allowed provided that it satisfies the martingale constraint.

Besides, using the martingale constraint $E[Y_i|X] = X_i$, we have that the cost is given by $$ E[(X_1-Y_1)(X_2-Y_2)] = E[Y_1 Y_2] - E[X_1 X_2] $$ which is independent of the joint law of $(X,Y)$. Hence the minimization is purely on the law of $X$ provided that the martingale constraint is satisfied.

I assume that what you mean by 'a solution' refers to the joint law, as it is usually the case with optimal transport problems. Since the value of the cost is independent of the joint law, the question does not make sense anymore.

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  • $\begingroup$ But the law of $X$ is not given and you need to consider all $X$ such that $(X,Y)$ form a martingale. The value does depend on the joint law. $\endgroup$
    – Ryan
    Commented May 14, 2018 at 22:13
  • $\begingroup$ The solution you are looking for is an $R^2$ valued random variable satisfying the martingale constraint, since the computation I wrote above shows that you can compute the cost you gave independently of the joint law. So you are not maximizing on a joint law with fixed marginals here, but rather on a random variable $X$ with some constraint on the joint law of $(X,Y)$. Once this constraint is satisfied, the value function does not depend on the joint law. Besides, since the first marginal is not fixed, you are not transporting anything. $\endgroup$
    – me47
    Commented May 14, 2018 at 22:55
  • $\begingroup$ i am not saying this is an standard optimal transport problem( maximizing cost over joint law with fixed marginals), for otherwise i wouldn't be asking the question. I am looking for joint law such that its second marginal is fixed, while its first marginal is free but needs to be able to coupled with the second one to form a martingale. If you look at the problem as a 2-periods problem, this is exactly the martingale optimal transport, where marginals at time 0 and time 2 are fixed, and marginal at time 1 is free. $\endgroup$
    – Ryan
    Commented May 15, 2018 at 15:58
  • $\begingroup$ what you pointed out has nothing to do with what i am asking. but thanks anyway. $\endgroup$
    – Ryan
    Commented May 15, 2018 at 16:02
  • $\begingroup$ By standard optimal transport I am including the martingale version. Again, the question you ask is not a martingale optimal transport question because the law of X is not fixed and this is crucial to apply all the results from this theory. If it were fixed, then it is part of the theory but, in your specific case the cost function does not depend on the joint law anyway. Is your question 'given a random variable Y, can I find any random variable X such that the joint law satisfies the martingale constraint?' In that case the answer is yes (provided mild assumptions on Y). $\endgroup$
    – me47
    Commented May 15, 2018 at 16:59

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