Convergence of stochastic process - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:35:10Z http://mathoverflow.net/feeds/question/93218 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93218/convergence-of-stochastic-process Convergence of stochastic process John Young 2012-04-05T14:19:48Z 2012-04-05T14:47:13Z <p>As we know, to prove the convergence of stochastic process, we could either show the convergence of finite dimensional distribution and tightness of the process, or use techniques of martingale problems. What about the following Markov process:</p> <p>$L=\frac{1}{2}p(1-p)\frac{d^{2}}{dp^{2}}-\frac{\theta}{2}p\frac{d}{dp}+\log(\theta) p(1-p)(2p-1)\frac{d}{dp}, p\in[0,1]$</p> <p>We can see that the generator explodes when $\theta\rightarrow0$. How can we find the limit of this process as $\theta\rightarrow0$. Apparently, the techniques of martingale problems are not applicable here! </p> http://mathoverflow.net/questions/93218/convergence-of-stochastic-process/93222#93222 Answer by Pascal Maillard for Convergence of stochastic process Pascal Maillard 2012-04-05T14:47:13Z 2012-04-05T14:47:13Z <p>This drifted Wright--Fisher diffusion seems to converge to the rather degenerate process $X_0 = p$, $X_t = 1/2$ for $t>0$, which is why I would do it by hand: Show that for every $t>0$ and every $p\in(0,1)$, the process started from $p$ is with high probably near $1/2$ at time $t$, uniformly for $p\in[\epsilon,1-\epsilon]$, say (you can use standard arguments from diffusion theory for that, i.e. speed measure, Green function...). This yields convergence in finite-dimensional distributions. Convergence in Skorokhod topology will not hold, since continuous functions are closed in this topology, hence a continuous process cannot converge to a process with a jump.</p>