Hello 
It is fun for me to sit back and watch some of the fascinating discussions that happen in the #autodiff channel on Slack. There is definitely some awesome brainpower going into this field of research.
I finished my PhD more than 16 years ago (UIUC 2002) and have been doing mostly nontechnical work in financial services since then, but I try my best to keep one foot in some research to keep my sanity.
Automatic differentiation is a beautiful subject. I was aware of it during graduate school and I could see how it related to what I was working on, but I never had a chance do any serious research on the subject, but have definitely admired it from afar.
A little over a week ago, I pointed out (on Slack) that you could use automatic differentiation for finite differences, i.e. something like “automatic finite difference” and was asking if anyone had worked on that. It might sound odd to use AD for FD, but sometimes your underlying space really is finite and a finite difference is the more appropriate model. The subject of my dissertation was differential geometry on finitary spaces so I have thought about this a little bit 
Using the same principles, I pointed out that AD could be used for stochastic differentiation. Although the maths is very similar, I figured the stochastic version would be more interesting to more people, so I wrote up a simple proof of concept in the package:
This package is, by no means, intended to be used by anyone in its current state. It is really just a proof of concept and it does that. It actually works 
Writing the package was the easy part, but to document it was trickier. There always seemed to me to be a bit of magic around AD. It is unreasonably effective.
So I looked at my code. Scribbled some notes. And iterated until finally a picture emerged that I was happy with. I spent most of the last 7 days condensing the notation and clarifying things to the point that now, in hindsight, it all seems completely natural and obvious 
Anyway, I had great fun and wrote up the details in a blog post (with creative name):
If you are a researcher in this area, it might be a big YAWN for you, but it was fun to write up and it leads to all kinds of avenues for future (or past) work. I say “or past” because I didn’t do any kind of literature search so for all I know this might all be old news from decades ago and obvious to everyone. If that is the case, I don’t mind. It was just a fun exercise for my aging brain so no harm done
In any case, I wanted to share it in case anyone here finds it interesting. I will probably continue to dabble with AD and write up results here and there, but I feel I am decades behind 
Cheers 
