Pete- "actuarial experience" means "you wasted time working for a living rather than being a student, so you aren't sufficiently dedicated to the study of mathematics." I'm not saying that I agree with this attitude, but it's certainly an attitude that I've seen expressed in some quarters.

It's worth commenting that all bets are off if the population has a distribution that doesn't have a finite variance.

Let me just add that I have seen this term used elsewhere within the literature on local search algorithms, although I don't have any references handy.

If you aren't familiar with it, check out the antithetic variate technique, which is an extremely simplified version of what you've done here. The idea in AV is to run the simulation first using U(0,1) independent random numbers U(1), U(2), ..., and then repeat the simulation using 1-U(1), 1-U(2), ... In many practical situations you generate U(0,1) random numbers and then transform them in some way to get random numbers according to a more complicated distribution. Unfortunately, this can often break the antithetic variate technique by ruining the negative correlation.

I'd expect that single precision would be quite accurate enough for your purposes and would run somewhat faster than double precision, particularly if you're using a GPU to do this.

First, you might consider using something more sophisticated than a piecewise linear function. Why not cubic splines instead? There's a huge literature on heuristics for locating knots for spline approximation- you could simply use one of these heuristics to get a reasonable set of knots, check to see whether the L2 error is sufficiently small and if not, then you could add more knots until it is sufficient.

I'm afraid that my answer would start with "take an introductory real analysis course, an introductory course in systems of ODE's, and an introductory course in optimization", so my advice is probably not going to be helpful to you. My background is in mathematics (and computer science) followed by graduate study in optimization, where there's a completely different collection of material that everyone is expected to know. I'd suggest that you look at books and lecture notes that have been written for engineers interested in learning about this stuff.