# compactness of a probability set

I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of martingale measures $\mathbb{P}$ on $\Omega$, i.e.

$$E^{\mathbb{P}}[X_t|\mathcal{F}_s]=X_s,\ \ \forall 0\leq s<t\leq 1$$

Here $X_t(\omega)=\omega_t$ is the canonical process and $\{\mathcal{F}_t\}$ is the natural filtration. Now we are given a sequence of probability distribution on $\mathbb{R}$: $\mu=(\mu_t)_{0\leq t\leq 1}$ and we define

$$\mathcal{M}_{\Omega}(\mu)=\{\mathbb{P}\in\mathcal{M}_{\Omega}: X_t \stackrel{\mathbb{P}}{\sim}\mu_t\}$$

In other words, $X_t$ has the marginal distribution $\mu_t$ under $\mathbb{P}$. Certainly we impose some conditions on $\mu$ such that $\mathcal{M}_{\Omega}(\mu)$ is not empty. My question is whether we can prove that $\mathcal{M}_{\Omega}(\mu)$ is compact. If yes, can we find some simple condtions on $\mu$ such that $\mathcal{M}_{\Omega}(\mu)$ is compact; if not, can we find a counterexample? Many thanks for your reply.

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Compact in the weak topology, I presume? –  Nate Eldredge Nov 25 '13 at 19:51
Yes, we equip $\mathcal{M}_{\Omega}(\mu)$ with the weak convergence –  CodeGolf Nov 25 '13 at 19:55
Is there an obvious example where $\mathcal{M}_\Omega(\mu)$ contains more than one element? –  Nate Eldredge Nov 25 '13 at 19:55
We wish to show tightness of $\mathcal M_{\Omega}(\mu)$. By Doob's submartingale inequality, we can establish uniform boundedness of the trajectories in probability using only the marginal at time $t = 1$. Then if trajectories are Hölder continuous almost surely, i.e. $|X(t) - X(s)| \leq M |t-s|^{\alpha}$, for some fixed constants $M$ and $0 < \alpha \leq 1$, we have by the Arzelà-Ascoli theorem that uniformly bounded sets of $\alpha$-Hölder continuous functions are relatively compact. The question then is whether we can show this kind of regularity (uniformly in probability is sufficient). –  Joris Bierkens Nov 25 '13 at 22:52
@Nate Eldredge: in the introduction (page xxviii) of the reference CodeGolf mentions, it is stated that for a given family of marginals $(\mu_t)$, under some conditions uniqueness of the associated martingale can be shown, but not in general. In fact the authors state that they provide numerous examples within the book where uniqueness does not hold (but I did not look so close as to find these examples). A reference for the uniqueness result is G. Lowther, "Fitting martingales to given marginales", 2008. –  Joris Bierkens Nov 25 '13 at 22:57