Question

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:

$L=\frac{1}{2}p(1-p)\frac{d^{2}}{dp^{2}}-\frac{\theta}{2}p\frac{d}{dp}+\log(\theta) p(1-p)(2p-1)\frac{d}{dp}, p\in[0,1]$

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!

Answer 1

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.