Markov Processes with Given Marginals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:38:09Z http://mathoverflow.net/feeds/question/87948 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87948/markov-processes-with-given-marginals Markov Processes with Given Marginals Tom Alberts 2012-02-09T01:23:36Z 2012-02-09T11:09:48Z <p>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:</p> <p>Is there a Markov process $X_t$ such that its marginal distribution at each time is $\mu_t$?</p> <p>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.</p> <h2>Edit</h2> <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/87948/markov-processes-with-given-marginals/87956#87956 Answer by Byron Schmuland for Markov Processes with Given Marginals Byron Schmuland 2012-02-09T03:24:58Z 2012-02-09T03:24:58Z <p>If you are willing to drop continuity in the parameter $t$, then you could let $(X_t)$ be independent with distribution $\mu_t$. </p> http://mathoverflow.net/questions/87948/markov-processes-with-given-marginals/87977#87977 Answer by The Bridge for Markov Processes with Given Marginals The Bridge 2012-02-09T10:10:28Z 2012-02-09T11:09:48Z <p>Hi, </p> <p>This is not the general answer you are looking for but it might be sufficient for your needs, here are my two cents. </p> <p>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. </p> <p>Take a look <a href="http://arxiv.org/abs/0910.3992" rel="nofollow">here</a>, 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. </p> <p>Best Regards</p>