Question

Let $\mu_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue measure. The basic question is as follows:

Is there a Markov process $X_t$ such that its marginal distribution at each time is $\mu_t$?

An obvious example is when $$d \mu_t = \frac{e^{-x^2/2t}}{\sqrt{2 \pi t}} dx$$ and $\mu_0 = \delta_0$, in which case we know that Brownian motion is such a Markov process. I am curious to know if there is any general theory along these lines.

Edit

As per Byron's comment below, I would like the Markov process to be continuous. Ideally I would like to have an SDE description of the process.

The SDE description actually suggests one possible answer: simply compute and play with the time and space derivatives of the density function to see if they satisfy some sort of parabolic equation (like the heat equation), use this to get the adjoint of the generator, and then compute the generator itself. This is a very plausible option, but I was hoping that there might be something more systematic.

Answer 1

If you are willing to drop continuity in the parameter $t$, then you could let $(X_t)$ be independent with distribution $\mu_t$.

Answer 2

Hi,

This is not the general answer you are looking for but it might be sufficient for your needs, here are my two cents.

If you are given a Semimartingale then there exists conditions that ensures that there exists a Markov process such that this markov process has the same marginal distributions that the marginal distributions of your semimartingale.

Take a look here, where it is motivated by application to mathematical finance and where the first version of this theorem is known as "Gyongy's Lemma" I believe.

Best Regards