> 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.