...and if my bargaining skills are not at the top of their game, a higher price thanI bargained for, too!

If you can point me to a resource that explains how to use this approach to solve my problem, I'll read up so I can consider accepting this answer...

Well, as per above, we're good on the locally linear model for the hidden variable. But the whole point is to discuss a highly non-normal model for the noise on the observation process.

I'm not sure I see how there is a difference other than the shape of the error distribution. The true state variable is p. I measure p, but i get back a noisy reading. The mean of my reading is p. But the distribution around p is not normal (it is bimodal). So I see this as a potential issue with distribution shape, rather than a broader philosophical difference. No?

Put another way, my observations are not error-free...just they are distributed "very-non-normally" around their true value.

I don't agree with the statement that I have no measurement noise. I flip a coin. At that instant, it has a true probability p of coming up heads (which I do not know). It comes up heads. That doesn't mean p=1.0. The outcome of the single coin toss should be assumed to be a draw from a probability distribution whose parameters are unknown and which we are trying to infer.

Thanks again for your time and consideration. I'd like to learn more about this topic, what's a good first text? Specifically I'd like to learn about the techniques for answering questions like the one above...when will a process specified by a particular SDE/SPDE have various properties, etc. I'm less concerned with the theoretical foundations (though I don't mind them) than I am with understanding how to answer specific questions for concrete examples.

Thanks for taking the time to explain. So let's talk only about the $\mu=0$ case: why is it that the probability of hitting 0 (in finite time) is 0 for $\beta=1$ but non-zero for any $\beta<1$? What's the logic for/proof for the fact that the cases split at $\beta=1$ rather than, say $\beta=1/2$ or $\beta=2/3$ or other?

I misstated the problem in my original post: the drift term should have the same scaling that the stochastic term does. Will fix above.