What do you use SDEs for? I always skipped diff eq topics because I never had a sensible application for them in the social sciences.
We also use the parameters for artificial pancreas systems, in order to give personalized and optimized insulin dosages for each patient, using those same exact parameters. In this case, we use a controller (model predictive control) and plug in the individualized parameters into the controller interface for each individual patient. The system is interfaced with a mobile device, an insulin pump, and a continuous glucose monitor. The mobile device sends data from the insulin pump and continuous glucose monitor to the controller (input) and retrieves data from the controller (output) via the mobile device. Via the output, commands are issued to the insulin pump regarding insulin dosing.
> The idea of time-reversing Brownian motion is like saying that you're running thermodynamics backward--it only makes sense if you sample them forward, then move backwards along the forward sampled motion.
You are conceptually 100% correct on this matter.
...As for me, I usually solve SDEs using Markov chain Monte Carlo, using esoteric software known as WinBUGS/OpenBUGS/MultiBUGS. I use MultiBUGS [1] (don't bother installing it on Windows or Mac), which is integrated with BlackBox Component Builder [2]. It all runs on a really cool but esoteric language called Component Pascal. Anyways, developers over the years have kept BlackBox in particular alive. But, I always have to say, differential equations are awesome :-) But, Julia is where it is at for DiffEqs in open source.
[1] https://www.multibugs.org/
[2] https://blackbox.oberon.org/